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YUFEI ZHAO: OK,
let's, get started.

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Welcome to 18.217.

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So this is combinatorial
theory, graph theory,

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and additive combinatorics.

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So course website is up there.

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So all the course
information is on there.

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So after around the
middle of the class,

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I'll say a bit more about
various course information,

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administrative things.

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But I want to jump directly
into the mathematical content.

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So this course
roughly has two parts.

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The first part will
look at graph theory,

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in particular problems
in extremal graph theory.

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In the second part,
we'll transition

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to additive combinatorics.

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But these are not two
separate subjects.

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So I want to show you
this topic in a way that

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connects these two
areas and show you

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that they are quite
related to each other.

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And many of the
common themes that

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will come up in one
part of the course

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will also show up in the other.

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So the story between
graph theory and additive

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combinatorics began
about 100 years ago

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with Schur, the famous
mathematician, Isaai Schur.

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Well, he was like
many mathematicians

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of his era trying to prove
Fermat's Last Theorem.

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So here's what's
Schur's approach.

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He said, well, let's look at
this equation that comes up

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in from Fermat's Last Theorem.

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And, well, one of the methods
of elementary number theory

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to rule out solutions
to an equation

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is to consider what
happens when you mod p.

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If you can rule out for
infinitely many values

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p, possible non-trivial
solutions to this equation mod

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p, then you will rule out
possibilities of solutions

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to Fermat's Last Theorem.

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OK, so this was
Schur's approach.

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As you can guess, unfortunately,
this approach did not work.

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And Schur proved that this
method definitely doesn't work.

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So that's the starting
point of our discussion.

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So it turns out that
for every value of n,

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there exists
non-trivial solutions

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for all p sufficiently large.

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So thereby, ruling
out the strategy.

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So let's see how Schur
proved his theorem.

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So that will be the first
half of today's lecture.

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So this seems like a
number theory question.

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So what does it have to
do with graph theory?

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So I wanted to show
you this connection.

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Now, Schur deduced his
theorem from another result.

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That is known as
Schur's Theorem, which

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says that if be
positive integers

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is colored using
finitely many colors,

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then there exists a
monochromatic solution

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to the equation x
plus y equals to z.

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So if you give me
10 colors and color

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the positive integers
using those 10 colors,

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then I can find
for you a solution

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to this equation where x, y,
and z are all of the same color.

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Now, this statement--

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OK, so it's a perfectly
understandable statement.

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But let me rephrase it in
a somewhat different way.

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And this gets to a
point that I want

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to discuss where many statements
in additive combinatorics

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or just combinatorics in general
have different formulations,

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one that comes in an
infinitary form, which

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is more qualitative so to
speak and another form that

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is known as finitary.

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And that's more
quantitative in nature.

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So Schur's Theorem is
stated in a infinitary form.

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So it tells you if you color
using finitely many colors,

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then there exists a
monochromatic solution.

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So many, but not all,
statements of that form

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have an equivalent finitary form
that is sometimes more useful.

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And also, once you stay
the right finitary form,

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you can ask
additional questions.

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So here's what Schur's Theorem
looks like in the equivalent

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finitary form.

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You give me an r.

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For every r, there exists
some N as a function of r,

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such that if the
numbers 1 through N--

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so throughout this course, I'm
going to use this bracket N

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to denote integers up to N--

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so if these numbers are
colored using our colors,

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then necessarily, there exists
a monochromatic solution

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to the equation x
plus y equals to z,

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where x, y, and z are in the
set that is being colored.

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So it looks very similar to
the first version I stated.

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But now, there are
some more quantifiers.

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So for every r,
there exists an N.

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So why are these two versions
equivalent to each other?

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So it's not too hard to
deduce their equivalence.

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So let me do that now.

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The fact that the
finitary version

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implies the infinitary
version claims

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should be fairly obvious.

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So once you know the
finitary version,

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if you give me a coloring of
the positive integers, well

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I just have to look
far enough up to this N

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and I get the conclusion I want.

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But now, in the
other direction--

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so in the other direction--

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suppose I fix to this r.

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So, OK, so I assume
the infinitary version.

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I wanted to deduce
the finitary version.

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So I start with this r.

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And let's suppose the
conclusion were false.

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So supposed the
conclusion were false,

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namely for every N there
exists some coloring--

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so for every N there
exists some coloring--

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which we will call
phi sub N, that

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avoids monochromatic solutions
to x plus y equals to z.

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So I'm going to use this Chi
for shorthand for monochromatic.

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So suppose there
exists such a coloring.

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And now, I want to take
this collection of colorings

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and produce for you a coloring
of the positive integers.

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And you can do this basically
by a standard diagonalization

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trick.

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Namely, we see that by taking
an infinite subsequent,

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such that--

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so let me call this infinite
sub-sequence phi of--

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phi sub-- well, so it's infinite
sub-sequence of this phi sub N,

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such that phi sub N
of k stabilizes along

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the sub-sequence for every k.

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OK, so you can do this simply
by diagonalization trick.

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And then, we see that
along the sub-sequence phi

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N converges point-wise to some
coloring of the entire set

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of positive integers.

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And this coloring avoids
monochromatic solutions

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to x plus y equals to
z, because if there

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were monochromatic
solutions in this coloring

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of the entire integers,
then I can look back

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to where that came from.

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And that would have been the
monochromatic solution in one

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of my phi N's.

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So this is an
argument that shows

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the equivalence between the
finitary form and infinitary

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form.

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But now, when we look
at the finitary form,

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you can ask additional
questions, such as,

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how big does this N have
to be as a function of r.

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It turns out those kind
of questions in general

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are very difficult. And
we know some things.

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For this type of questions,
we know some bounds usually.

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But the truth is
usually unknown.

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And there are
major open problems

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in combinatorics of this type.

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So there's still a lot
that we do not understand.

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OK, so now we have Schur's
Theorem in this form.

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Let me show you how to deduce
his conclusion about ruling out

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this approach to proving
Fermat's Last theorem.

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The claim is the following that
if you have a positive integer

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n, then for all
sufficiently large primes p,

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there exists x, y,
and z, all belonging

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to integers up to p minus 1,
such that their n-th powers

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add up like this.

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So it's a solution to
Fermat's equation mod p.

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All right, so how
can we deduce this

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from what we said
about coloring?

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So what is the coloring?

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OK, so here's what Schur
did, so proof assuming

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for now Schur's theorem.

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So let's look at the
multiplicative group

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of non-zero residues, mod p.

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So we know it's a cyclic group
because there's a generator.

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So there's a primitive
root generator.

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Let H denote the
subgroup of n-th powers.

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Well, H is a pretty
big subgroup.

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So what's the index of H in
this multiplicative group?

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It's at most M.

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So think about representing
this as a cyclic group using

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a generator.

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So H then would be all the
elements whose exponent

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is divisible by M. So
this the index is at most

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M. It could be smaller.

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But it's at most M.

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And so in particular,
I can use the H cosets

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to partition the multiplicative
group of non-zero residues.

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And this is a color.

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Virtual partition is the
same thing as a coloring.

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There is a bounded
number of colors.

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But I let peek at large.

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So by Schur's theorem if
p is sufficiently large,

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then one of my cosets
should contain a solution

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to x plus y equals to z.

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What does that look like?

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So that one coset, one H coset,
course that contains x, y, z,

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such that x plus y
equals to z as integers.

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They belong to the same coset.

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So x, y, and z belong
to some coset of H,

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which means then that x equals
to a times n-th power with a y

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equals to a times some
n-th power and little z

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equals to a times
some n-th power.

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You have this equation.

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Put them together.

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So that is true.

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So now mod p, I
can cancel the a's.

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And this produces a
non-trivial solution

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to Fermat's equation, mod p.

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OK, so this was the
proof of this claim

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that this method does
not work for solving

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Fermat's Last Theorem.

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But, you know, we assumed
this claim of Schur's theorem

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that every finite coloring
of the positive integers

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contains a monochromatic
solution to x plus y

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equals to z.

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So we still need to
prove that claim.

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So we still need to prove
this combinatorial claim.

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And so that's what
we're going to do now.

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This is where graph
theory comes in.

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So let me state a very
similar-looking theorem

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about graphs.

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And this is known
as Ramsey's theorem,

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although Ramsay's theorem
actually historically came

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after Schur's theorem, but
Ramsey's theorem, here, we're

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going to use it specifically
in the case for triangles.

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So what does it say?

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That if you give me an
r, the number of colors,

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then there exists
some large N such

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that if the edges of the
complete graph, K sub N,

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along N vertices are
colored using r colors,

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then there exists a
monochromatic triangle

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somewhere.

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Any questions so far about
any of these statements?

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So let's see how Ramsay's
theorem for triangles

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is proved.

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By the way, I want to give you
a historical note about Frank

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Ramsey.

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So he's someone who made
significant contributions

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to many different areas,
not just in mathematics.

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So he contributed to seminal
works in mathematical logic

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where this theorem came
from, but also to philosophy

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and to economics before his
untimely death at the age of 26

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from liver-related problems.

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So he's someone whose very short
life contributed tremendously

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to academics.

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So let's see how Ramsay's
theorem, in this case,

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is proved.

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00:19:47,780 --> 00:19:52,830
We'll do induction on
r, the number of colors.

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So for every r, I need
to show you some N,

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such that the statement is true.

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00:20:01,430 --> 00:20:07,040
In the first case, when r equals
to 1, there's not much to do.

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Just one color, if I
just have three vertices,

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that already is OK.

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00:20:12,830 --> 00:20:16,220
Three vertices, that's already
a monochromatic triangle.

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00:20:16,220 --> 00:20:21,390
So from now on, let
r be at least 2.

248
00:20:21,390 --> 00:20:35,480
And suppose the claim
holds for r minus 1 colors,

249
00:20:35,480 --> 00:20:42,440
with N prime being the
corresponding number

250
00:20:42,440 --> 00:20:46,280
of vertices with
r minus 1 colors.

251
00:20:46,280 --> 00:20:48,185
So now let me pick
an arbitrary vertex.

252
00:20:53,260 --> 00:20:57,290
So pick an arbitrary vertex
v and look what happens.

253
00:20:57,290 --> 00:20:59,810
So here's v.

254
00:20:59,810 --> 00:21:03,620
And let me look at
the outgoing edges.

255
00:21:09,940 --> 00:21:16,930
So we'll show that
N being r crimes

256
00:21:16,930 --> 00:21:21,650
N prime minus 1 plus 2 works.

257
00:21:21,650 --> 00:21:26,260
So now, we have a lot
of outgoing edges.

258
00:21:26,260 --> 00:21:31,120
In particular, we
have r times N prime

259
00:21:31,120 --> 00:21:35,530
minus 1 plus 1 outgoing edges.

260
00:21:35,530 --> 00:21:45,820
So by the pigeonhole
principle, some color--

261
00:21:45,820 --> 00:21:56,180
so there exists at least
N prime outgoing edges

262
00:21:56,180 --> 00:22:05,110
with the same color,
let's say, yellow.

263
00:22:05,110 --> 00:22:07,370
So suppose yellow is
the outgoing color.

264
00:22:14,260 --> 00:22:16,420
And let me call
the set of vertices

265
00:22:16,420 --> 00:22:18,460
on the other end
of these edges v0.

266
00:22:22,520 --> 00:22:24,736
So now let's think about
what happens in v0.

267
00:22:24,736 --> 00:22:41,020
So in v0, either v0 contains
a yellow edge, in which case

268
00:22:41,020 --> 00:22:42,880
you get a yellow triangle.

269
00:22:53,002 --> 00:22:56,163
Or we lose the color inside v0.

270
00:22:56,163 --> 00:22:57,580
So the number of
colors goes down.

271
00:23:00,835 --> 00:23:11,680
Else v0 has at most
r minus 1 colors.

272
00:23:11,680 --> 00:23:22,090
And v0 has at least N
prime number of vertices.

273
00:23:22,090 --> 00:23:32,600
So by induction, v0 has
a monochromatic triangle

274
00:23:32,600 --> 00:23:34,030
in the remaining colors.

275
00:23:40,180 --> 00:23:42,880
So that completes the proof of
Ramsay's theorem, in this case,

276
00:23:42,880 --> 00:23:44,860
for triangles.

277
00:23:44,860 --> 00:23:49,337
And if you wish to find out
what is the bound that comes out

278
00:23:49,337 --> 00:23:51,420
of this argument, well,
you can chase to the proof

279
00:23:51,420 --> 00:23:52,212
and get some bound.

280
00:23:55,098 --> 00:23:57,140
The remaining question
now is, what does this all

281
00:23:57,140 --> 00:23:59,990
have to do with Schur's theorem?

282
00:23:59,990 --> 00:24:02,760
So so far, we've talked
about some number theory.

283
00:24:02,760 --> 00:24:04,320
We've talked about
some graph theory

284
00:24:04,320 --> 00:24:07,350
and how to link these
two things together.

285
00:24:07,350 --> 00:24:08,850
And I think this
is a great example.

286
00:24:08,850 --> 00:24:10,520
It's a fairly simple
example, which

287
00:24:10,520 --> 00:24:14,240
I'm about to show you of how to
link these two ideas together.

288
00:24:14,240 --> 00:24:16,985
And this connection,
we'll see many times

289
00:24:16,985 --> 00:24:18,110
in the rest of this course.

290
00:24:28,820 --> 00:24:30,650
I don't want to erase
Schur's theorem.

291
00:24:30,650 --> 00:24:31,390
So let me--

292
00:24:47,570 --> 00:24:50,510
So let's prove Schur's theorem.

293
00:24:57,820 --> 00:24:59,190
So let's start with a coloring.

294
00:25:10,174 --> 00:25:14,130
So let's start with the
coloring of 1 through N.

295
00:25:14,130 --> 00:25:18,510
And I want to form
a graph with colors

296
00:25:18,510 --> 00:25:21,090
on the edges that
are somehow derived

297
00:25:21,090 --> 00:25:24,210
from this coloring
on these integers.

298
00:25:24,210 --> 00:25:26,070
And here's what I'm going to do.

299
00:25:26,070 --> 00:25:30,110
So let's color the
complete graph-- let's

300
00:25:30,110 --> 00:25:46,320
color the edges of the complete
graph on the vertex set

301
00:25:46,320 --> 00:25:49,470
having N plus 1 vertices,
labeled at integers

302
00:25:49,470 --> 00:25:54,160
up to positive integers
up to N plus 1.

303
00:25:54,160 --> 00:26:08,060
But by the Ramsey result we just
proved, if N is large enough,

304
00:26:08,060 --> 00:26:09,910
then there exists a
monochromatic triangle.

305
00:26:21,867 --> 00:26:22,950
So what does it look like?

306
00:26:26,010 --> 00:26:30,900
So let me draw for you a
monochromatic triangle.

307
00:26:30,900 --> 00:26:35,280
Suppose it-- so I haven't told
you what the coloring is yet.

308
00:26:35,280 --> 00:26:37,290
So the coloring
is that I'm going

309
00:26:37,290 --> 00:26:43,090
to color the edge
between i and j,

310
00:26:43,090 --> 00:26:48,460
using the color derived by
applying phi to the number

311
00:26:48,460 --> 00:26:52,150
j minus i, namely the
length of that segment

312
00:26:52,150 --> 00:26:56,800
if I lay out all the
vertices on the number line.

313
00:26:56,800 --> 00:27:00,550
So now have an r coloring
of this complete graph.

314
00:27:00,550 --> 00:27:03,010
So Ramsey tells us
that there exists

315
00:27:03,010 --> 00:27:05,260
a monochromatic triangle.

316
00:27:05,260 --> 00:27:10,510
The triangle sits on
vertices i, j, and k.

317
00:27:10,510 --> 00:27:18,670
And the rule tells us that the
colors are phi of k minus i,

318
00:27:18,670 --> 00:27:23,470
phi of j minus i,
and phi of k minus j.

319
00:27:30,810 --> 00:27:33,660
So these three numbers,
they have the same coloring.

320
00:27:33,660 --> 00:27:41,060
But, look, if I set these
numbers to be x, y, and z--

321
00:27:41,060 --> 00:27:44,540
so x being j minus
i, for instance--

322
00:27:44,540 --> 00:27:49,970
then x plus y equals to z.

323
00:27:49,970 --> 00:27:53,960
And they all have
the same color.

324
00:27:57,040 --> 00:28:00,420
So this monochromatic
triangle gives us

325
00:28:00,420 --> 00:28:03,990
a monochromatic
equation, 2x plus y

326
00:28:03,990 --> 00:28:11,810
equals to z, thereby concluding
the proof of Schur's theorem.

327
00:28:11,810 --> 00:28:15,660
OK, so this rounds out the
discussion for now of--

328
00:28:15,660 --> 00:28:21,090
well, we started with some
statement about number theory.

329
00:28:21,090 --> 00:28:25,960
And then we took this
detour to graph theory,

330
00:28:25,960 --> 00:28:29,920
looking at Ramsay's theorems
of monochromatic triangles,

331
00:28:29,920 --> 00:28:32,140
and then go back
to number theory

332
00:28:32,140 --> 00:28:36,140
and proved the result
that Schur did.

333
00:28:36,140 --> 00:28:40,260
So how does go to graphs help?

334
00:28:40,260 --> 00:28:41,940
So why was this advantageous?

335
00:28:45,642 --> 00:28:46,600
What do you guys think?

336
00:28:54,290 --> 00:28:58,530
So I claim that by
going to graphs,

337
00:28:58,530 --> 00:29:04,710
we added some extra flexibility
to what we can play with.

338
00:29:04,710 --> 00:29:07,620
For example, we started
out with a problem

339
00:29:07,620 --> 00:29:12,370
where there were only
N things being colored.

340
00:29:12,370 --> 00:29:15,670
And then we moved to
graphs where about--

341
00:29:15,670 --> 00:29:19,510
well, N choose 2 or N squared
objects are being colored.

342
00:29:19,510 --> 00:29:21,840
And then we did an
induction argument.

343
00:29:21,840 --> 00:29:25,570
So remember in the proof of
Ramsey's theorem up there,

344
00:29:25,570 --> 00:29:28,900
there was an induction
argument taking all vertices.

345
00:29:28,900 --> 00:29:33,340
And that argument doesn't
make that much sense

346
00:29:33,340 --> 00:29:35,155
if you stayed
within the numbers.

347
00:29:37,790 --> 00:29:39,620
Somehow moving to
graphs gave you

348
00:29:39,620 --> 00:29:43,310
that extra flexibility
allow you to do more things.

349
00:29:43,310 --> 00:29:45,440
And this is one
of the advantages

350
00:29:45,440 --> 00:29:49,480
of moving from
problem about numbers

351
00:29:49,480 --> 00:29:51,430
to a problem about graphs.

352
00:29:51,430 --> 00:29:54,030
And we'll see this
connection later on as well.

353
00:29:54,030 --> 00:29:55,090
Yeah?

354
00:29:55,090 --> 00:29:56,590
AUDIENCE: Sort of
related to that.

355
00:29:56,590 --> 00:30:00,590
Are there better bounds
known for this specific,

356
00:30:00,590 --> 00:30:04,340
like Schur's result
of that power on e,

357
00:30:04,340 --> 00:30:06,590
because the N's here
would be pretty bad.

358
00:30:06,590 --> 00:30:10,317
YUFEI ZHAO: Right, so Ashwan
asked, so what about bounds?

359
00:30:10,317 --> 00:30:11,650
So what do we know about bounds?

360
00:30:11,650 --> 00:30:14,298
So I don't know off the
top of my head the answers

361
00:30:14,298 --> 00:30:15,090
to those questions.

362
00:30:15,090 --> 00:30:17,940
But in general,
they're quite open.

363
00:30:17,940 --> 00:30:20,880
So there are exponential gaps
between lower and upper bounds

364
00:30:20,880 --> 00:30:23,122
on our knowledge of what
is the optimal N you

365
00:30:23,122 --> 00:30:24,080
can put in the theorem.

366
00:30:26,640 --> 00:30:28,057
Any more questions?

367
00:30:31,400 --> 00:30:35,280
All right so, I think this is
a good point for us to-- so

368
00:30:35,280 --> 00:30:37,770
usually when I give
90-minute lectures,

369
00:30:37,770 --> 00:30:40,920
I like to take a short
2-minute break in between.

370
00:30:40,920 --> 00:30:41,880
So I want to do that.

371
00:30:41,880 --> 00:30:43,350
And then in the
second half, I want

372
00:30:43,350 --> 00:30:47,940
to take you through a tour
of additive combinatorics.

373
00:30:47,940 --> 00:30:51,020
So tell you about some of
the modern developments.

374
00:30:51,020 --> 00:30:54,720
Now, this is an exciting
field where it started out,

375
00:30:54,720 --> 00:30:58,500
I think, roughly with Schur's
theorem that we just discussed.

376
00:30:58,500 --> 00:31:00,300
That started about
100 years ago.

377
00:31:00,300 --> 00:31:03,510
But a lot has taken place
in the past century.

378
00:31:03,510 --> 00:31:06,720
And there's still a lot of
ongoing exciting research

379
00:31:06,720 --> 00:31:07,470
developments.

380
00:31:07,470 --> 00:31:09,280
So in the second
half of this lecture,

381
00:31:09,280 --> 00:31:13,110
I want to give you a tour
through those developments

382
00:31:13,110 --> 00:31:14,970
and show you some
of the highlights

383
00:31:14,970 --> 00:31:16,428
from additive combinatorics.

384
00:31:16,428 --> 00:31:17,970
So let's take a
quick 2-minute break.

385
00:31:17,970 --> 00:31:22,506
And feel free to ask
questions in the meantime.

386
00:31:22,506 --> 00:31:24,280
So another part of
the writing assignment

387
00:31:24,280 --> 00:31:27,970
in addition to course
notes is a contribution

388
00:31:27,970 --> 00:31:32,198
to Wikipedia, which is, you
know, nowadays, of course,

389
00:31:32,198 --> 00:31:34,115
you know, if you hear
some word like Szemeredi

390
00:31:34,115 --> 00:31:36,280
's regularity lemma
the first thing you do

391
00:31:36,280 --> 00:31:37,570
is type into Google.

392
00:31:37,570 --> 00:31:39,760
And more often than not the
first link that comes up

393
00:31:39,760 --> 00:31:41,610
is Wikipedia.

394
00:31:41,610 --> 00:31:43,260
And, you know, some
of the articles,

395
00:31:43,260 --> 00:31:47,940
they are all right, and some of
them are really not all right.

396
00:31:47,940 --> 00:31:51,480
And it would be fantastic
for future students

397
00:31:51,480 --> 00:31:56,100
and also for yourselves
if there were better entry

398
00:31:56,100 --> 00:32:00,540
points to this area by having
higher quality Wikipedia

399
00:32:00,540 --> 00:32:03,330
articles or articles
that are simply

400
00:32:03,330 --> 00:32:05,530
missing about specific topics.

401
00:32:05,530 --> 00:32:07,145
So one of the assignments--

402
00:32:07,145 --> 00:32:08,520
again, this can
be collaborative.

403
00:32:08,520 --> 00:32:09,895
So I'll give you
more information

404
00:32:09,895 --> 00:32:11,010
how to do that later--

405
00:32:11,010 --> 00:32:14,340
is to contribute to
Wikipedia and roughly

406
00:32:14,340 --> 00:32:16,500
contribute one high
quality article

407
00:32:16,500 --> 00:32:18,540
or edit some
existing articles so

408
00:32:18,540 --> 00:32:21,280
that they become high quality.

409
00:32:21,280 --> 00:32:21,780
Yep.

410
00:32:21,780 --> 00:32:24,986
AUDIENCE: Can we
something similar to LMDB

411
00:32:24,986 --> 00:32:27,880
with creating a website
that has all the information

412
00:32:27,880 --> 00:32:30,007
needed in combinatorics?

413
00:32:30,007 --> 00:32:31,590
YUFEI ZHAO: So we
can talk about that.

414
00:32:31,590 --> 00:32:34,770
So if there are other
ideas about how to do this,

415
00:32:34,770 --> 00:32:37,363
we can definitely open
the chatting about that.

416
00:32:37,363 --> 00:32:38,780
So the other thing
is that instead

417
00:32:38,780 --> 00:32:42,680
of holding the usual office
hours, what I like to do is--

418
00:32:42,680 --> 00:32:45,170
so this class ends at 4:00 PM.

419
00:32:45,170 --> 00:32:47,330
So after 4:00, I'll go
up to the Math Common

420
00:32:47,330 --> 00:32:49,090
Room, which is
just right upstairs

421
00:32:49,090 --> 00:32:50,360
and hang out there for a bit.

422
00:32:50,360 --> 00:32:53,340
If you have questions, you
want to chat, come talk to me.

423
00:32:53,340 --> 00:32:57,050
I'd be happy to chat about
anything related or not related

424
00:32:57,050 --> 00:32:58,030
to the course.

425
00:32:58,030 --> 00:33:00,290
And before homeworks
are due, I will

426
00:33:00,290 --> 00:33:03,890
try to set up some special
office hours for you

427
00:33:03,890 --> 00:33:06,215
in case you want to ask
about homework problems.

428
00:33:06,215 --> 00:33:08,090
And if you want to meet
with me individually,

429
00:33:08,090 --> 00:33:09,970
please just send me an email.

430
00:33:09,970 --> 00:33:11,720
Oh, one more thing
about the course notes.

431
00:33:11,720 --> 00:33:16,290
So because I want to
do quality control,

432
00:33:16,290 --> 00:33:19,910
so here is the process that will
happen with the course notes.

433
00:33:19,910 --> 00:33:21,890
So the first lecture
is already online.

434
00:33:21,890 --> 00:33:22,940
So you can already see.

435
00:33:22,940 --> 00:33:25,790
So I've written up the lecture
notes for the first lecture.

436
00:33:25,790 --> 00:33:27,620
And you can use
that as an example

437
00:33:27,620 --> 00:33:30,210
of what I'm looking for.

438
00:33:30,210 --> 00:33:32,098
So I'm looking for
people to sign up

439
00:33:32,098 --> 00:33:33,640
starting from the
next lecture, and I

440
00:33:33,640 --> 00:33:36,470
will send out a link tonight.

441
00:33:36,470 --> 00:33:41,040
For future lectures, so
whoever writes the lecture,

442
00:33:41,040 --> 00:33:45,290
I'll the lecture, and
then within one day, so

443
00:33:45,290 --> 00:33:48,250
by the end of the day
after the lecture,

444
00:33:48,250 --> 00:33:51,080
it will be good if
they were already

445
00:33:51,080 --> 00:33:53,065
at least some sketch,
some rough draft at least

446
00:33:53,065 --> 00:33:54,440
containing the
theorem statements

447
00:33:54,440 --> 00:33:56,390
and whatnot from
the day's lecture.

448
00:33:56,390 --> 00:34:01,100
So that the next person can
start writing afterwards.

449
00:34:01,100 --> 00:34:03,290
But once you are
done, once you feel

450
00:34:03,290 --> 00:34:06,740
that you have a polished version
of the lecture, write up,

451
00:34:06,740 --> 00:34:09,810
ideally within four
days of the lecture--

452
00:34:09,810 --> 00:34:12,230
so that in terms of
expectations and timelines,

453
00:34:12,230 --> 00:34:14,480
again all of this
information is online--

454
00:34:14,480 --> 00:34:17,150
so you're finished with
polishing your lecture notes,

455
00:34:17,150 --> 00:34:20,120
within four days send me an
email, so both co-authors

456
00:34:20,120 --> 00:34:22,310
if there are two of
you, and I will schedule

457
00:34:22,310 --> 00:34:25,100
an appointment,
about half an hour,

458
00:34:25,100 --> 00:34:26,600
where I will sit
down with you to go

459
00:34:26,600 --> 00:34:29,630
through what you've written
and tell you some comments.

460
00:34:29,630 --> 00:34:31,790
So you can go back
and polish it further.

461
00:34:31,790 --> 00:34:34,699
And hopefully, that will
just be a one round thing.

462
00:34:34,699 --> 00:34:37,670
If more rounds are needed,
well, it's not ideal,

463
00:34:37,670 --> 00:34:42,199
but we'll make it happen until
the notes are ready to use

464
00:34:42,199 --> 00:34:44,429
for future generations.

465
00:34:44,429 --> 00:34:50,150
OK, any questions about any
of the course logistics?

466
00:34:50,150 --> 00:34:53,699
All right, so in the second
half of today's lecture,

467
00:34:53,699 --> 00:34:57,630
I want to take you through
a tour of modern additive

468
00:34:57,630 --> 00:34:58,830
combinatorics.

469
00:34:58,830 --> 00:35:01,320
And this is an area
of research which

470
00:35:01,320 --> 00:35:03,000
I am actively involved in.

471
00:35:03,000 --> 00:35:05,130
And it's something that
I am quite excited about.

472
00:35:05,130 --> 00:35:08,010
And part of the reason
why I teach this course--

473
00:35:08,010 --> 00:35:10,980
this course is something that
I developed a couple years ago

474
00:35:10,980 --> 00:35:12,810
when I taught for
the first time then--

475
00:35:12,810 --> 00:35:14,730
because I want to
introduce you guys

476
00:35:14,730 --> 00:35:20,370
to this very active and
exciting area of research.

477
00:35:20,370 --> 00:35:25,040
Now, what is added
combinatorics?

478
00:35:25,040 --> 00:35:27,360
The term itself is
actually fairly new.

479
00:35:27,360 --> 00:35:30,450
So the term, additive
combinatorics, I believe

480
00:35:30,450 --> 00:35:35,190
was coined by Terry Tao back
in the early 2000s as somewhat

481
00:35:35,190 --> 00:35:39,990
of a rebranding of an area that
already existed, but then got

482
00:35:39,990 --> 00:35:43,390
a lot of exciting developments
in the early 2000s.

483
00:35:43,390 --> 00:35:47,460
It's a deep and far reaching
subject with many connections

484
00:35:47,460 --> 00:35:51,600
to areas like graph theory,
harmonic analysis, or Fourier

485
00:35:51,600 --> 00:35:56,730
analysis, ergodic theory,
discrete geometry, logic

486
00:35:56,730 --> 00:36:00,540
and model theory, and has many
connections all over the place,

487
00:36:00,540 --> 00:36:03,000
and also has many deep theorems.

488
00:36:03,000 --> 00:36:05,880
So let me take you through a
tour historically of, I think,

489
00:36:05,880 --> 00:36:08,700
some of the major
milestones and landmarks

490
00:36:08,700 --> 00:36:12,702
in additive combinatorics.

491
00:36:12,702 --> 00:36:15,060
So after Schur's theorem,
which we discussed

492
00:36:15,060 --> 00:36:19,370
in the first half
of today's lecture,

493
00:36:19,370 --> 00:36:24,640
the next big result I would say
is Van der Waerden's theorem,

494
00:36:24,640 --> 00:36:30,210
which was 1927.

495
00:36:30,210 --> 00:36:36,390
Van der Waerden's theorem
says that every coloring

496
00:36:36,390 --> 00:36:49,610
of the positive integers
using finite many colors

497
00:36:49,610 --> 00:36:55,029
contains arbitrarily long
arithmetic progressions.

498
00:37:06,010 --> 00:37:09,050
So we'll see arithmetic
progressions come up a lot.

499
00:37:09,050 --> 00:37:12,730
So from now on we'll
abbreviate this word by AP.

500
00:37:12,730 --> 00:37:17,420
So AP stands for
Arithmetic Progressions.

501
00:37:17,420 --> 00:37:19,450
So instead of Schur's
theorem where you just

502
00:37:19,450 --> 00:37:22,070
find a single solution
to x plus y equals to z,

503
00:37:22,070 --> 00:37:26,020
so now, we're finding a
much bigger structure.

504
00:37:26,020 --> 00:37:28,750
Keep in mind, so a
novice mistake people

505
00:37:28,750 --> 00:37:32,440
make is to confuse arbitrarily
long arithmetic progressions

506
00:37:32,440 --> 00:37:33,580
with infinitely long.

507
00:37:33,580 --> 00:37:35,562
So these are definitely
not the same.

508
00:37:35,562 --> 00:37:36,520
So you can think about.

509
00:37:36,520 --> 00:37:39,310
I'll leave it to you as
an exercise, well, also

510
00:37:39,310 --> 00:37:43,270
homework exercise, that you can
color the integers with just

511
00:37:43,270 --> 00:37:45,820
two colors in a
way that destroys

512
00:37:45,820 --> 00:37:49,570
all possible infinitely long
monochromatic arithmetic

513
00:37:49,570 --> 00:37:51,130
progressions.

514
00:37:51,130 --> 00:37:53,980
So arbitrarily long is very
different from infinitely long.

515
00:37:53,980 --> 00:38:00,100
Now, so this was a great result,
but it provokes more questions.

516
00:38:00,100 --> 00:38:12,020
So Erdos-Turan in the
'30s, they asked--

517
00:38:12,020 --> 00:38:16,490
well, they conjectured
that the true reason in Van

518
00:38:16,490 --> 00:38:19,970
der Waerden's theorem of having
long arithmetic progressions,

519
00:38:19,970 --> 00:38:22,340
it's not so much
that you're coloring.

520
00:38:22,340 --> 00:38:25,550
It's just because if you
use finitely many colors,

521
00:38:25,550 --> 00:38:30,110
then one of the color classes
must have fairly high density.

522
00:38:30,110 --> 00:38:33,800
So one of the classes if you use
r colors has density at least 1

523
00:38:33,800 --> 00:38:34,940
over r.

524
00:38:34,940 --> 00:38:43,000
And they conjectured
that every subset

525
00:38:43,000 --> 00:38:54,400
of the positive integers, or the
integers with positive density,

526
00:38:54,400 --> 00:39:02,040
contains long--

527
00:39:02,040 --> 00:39:05,827
so arbitrarily long
arithmetic progressions.

528
00:39:08,373 --> 00:39:10,040
You may ask, what
does it mean, density?

529
00:39:10,040 --> 00:39:12,910
So you can define density
in many different ways.

530
00:39:12,910 --> 00:39:14,770
And it doesn't
actually really matter

531
00:39:14,770 --> 00:39:16,450
that much which
definition you use.

532
00:39:16,450 --> 00:39:19,280
But let me write
down one definition.

533
00:39:19,280 --> 00:39:24,100
So you can define given
a subset of integers

534
00:39:24,100 --> 00:39:27,100
the upper density, or
rather, let me just

535
00:39:27,100 --> 00:39:38,680
say that it has positive
upper density, if when we take

536
00:39:38,680 --> 00:39:44,450
the lim sup as n goes to
infinity and look at we'll

537
00:39:44,450 --> 00:39:48,980
take a scaling window and look
at what fraction of that window

538
00:39:48,980 --> 00:40:02,700
is a, then this number,
this limit sup is positive.

539
00:40:02,700 --> 00:40:05,340
So that's one definition
of positive density.

540
00:40:05,340 --> 00:40:08,075
There are many other
definitions, sometimes known

541
00:40:08,075 --> 00:40:09,470
as the Banach density.

542
00:40:09,470 --> 00:40:11,830
And you can take variations.

543
00:40:11,830 --> 00:40:13,680
I mean, for the purpose
of this discussion,

544
00:40:13,680 --> 00:40:15,350
they're all roughly equivalent.

545
00:40:15,350 --> 00:40:19,230
So let's not worry too
much about which definition

546
00:40:19,230 --> 00:40:22,150
of density we use here.

547
00:40:22,150 --> 00:40:24,550
All right, so Erdos
and Turan conjectured

548
00:40:24,550 --> 00:40:26,820
that the true reason for
Van der Waerden's theorem

549
00:40:26,820 --> 00:40:30,480
is that one of the color
classes has positive density.

550
00:40:30,480 --> 00:40:34,530
And this turned out to be an
amazingly prescient question

551
00:40:34,530 --> 00:40:39,120
and that one had to
wait several decades.

552
00:40:39,120 --> 00:40:43,710
So this conjecture was
made in the '30s, in 1936.

553
00:40:43,710 --> 00:40:46,350
So you had to wait several
decades before finding out

554
00:40:46,350 --> 00:40:48,100
what the answer is.

555
00:40:48,100 --> 00:40:52,870
So in a foundational
theorem, in the subject

556
00:40:52,870 --> 00:40:55,330
known as Roth's theorem--

557
00:40:55,330 --> 00:40:58,278
so Roth proved it in the '50s.

558
00:40:58,278 --> 00:40:58,820
I think '53--

559
00:41:13,730 --> 00:41:16,520
Roth proved that, I
think, '53, in the '50s--

560
00:41:16,520 --> 00:41:19,590
that k equals to 3 is true.

561
00:41:19,590 --> 00:41:25,330
So if I say that it
contains k term, arithmetic

562
00:41:25,330 --> 00:41:28,610
progressions for every k.

563
00:41:28,610 --> 00:41:31,180
And Roth proved that
every positive density

564
00:41:31,180 --> 00:41:33,340
subset contains a 3-term
arithmetic progression.

565
00:41:33,340 --> 00:41:37,160
And already, Roth introduced
very important ideas

566
00:41:37,160 --> 00:41:40,427
that we will see in this
course in two different forms.

567
00:41:40,427 --> 00:41:42,010
So in the first half
the course, we'll

568
00:41:42,010 --> 00:41:44,860
see a graph theoretic
proof that was found later

569
00:41:44,860 --> 00:41:47,433
in the '70s of Roth's theorem.

570
00:41:47,433 --> 00:41:48,850
And then in the
second half, we'll

571
00:41:48,850 --> 00:41:52,800
see Roth's original proof
that used Fourier analysis.

572
00:41:52,800 --> 00:41:55,220
So Fourier analysis
in number theory

573
00:41:55,220 --> 00:41:57,620
is also known as the
Hardy-Littlewood circle method.

574
00:41:57,620 --> 00:42:00,230
It's a powerful method in
analytic number theory.

575
00:42:00,230 --> 00:42:04,730
But there are very interesting
new ideas introduced by Roth as

576
00:42:04,730 --> 00:42:10,900
well in developing this result.

577
00:42:10,900 --> 00:42:14,615
The full conjecture was
settled by Szemeredi.

578
00:42:17,663 --> 00:42:19,080
It took another
couple of decades.

579
00:42:22,800 --> 00:42:29,980
So in the late '70s, Szemeredi
proved his landmark theorem

580
00:42:29,980 --> 00:42:34,330
that confirmed the
Erdos-Turan conjecture.

581
00:42:34,330 --> 00:42:36,600
Szemeredi's theorem
is a deep theorem.

582
00:42:36,600 --> 00:42:39,090
So this theorem
is the proof, what

583
00:42:39,090 --> 00:42:43,500
the original combinatorial
proof is a tour de force.

584
00:42:43,500 --> 00:42:46,290
And you can look
at the introduction

585
00:42:46,290 --> 00:42:51,600
of his paper, where there is
an enormously complex diagram--

586
00:42:51,600 --> 00:42:53,490
so you can see this
in the course notes--

587
00:42:53,490 --> 00:42:57,270
that lays out the logical
dependencies of all the lemmas

588
00:42:57,270 --> 00:42:59,040
and propositions in his paper.

589
00:42:59,040 --> 00:43:02,280
And even if you assume every
single statement is true,

590
00:43:02,280 --> 00:43:04,650
looking at that diagram,
it's not immediately clear

591
00:43:04,650 --> 00:43:07,170
what is going on because
the logical dependencies are

592
00:43:07,170 --> 00:43:08,740
so involved.

593
00:43:08,740 --> 00:43:14,320
So this was a really
complex proof.

594
00:43:14,320 --> 00:43:17,910
But not only that, Szemeredi's
theorem actually motivated

595
00:43:17,910 --> 00:43:20,030
a lot of subsequent research.

596
00:43:20,030 --> 00:43:23,700
So later on, researchers
from other areas

597
00:43:23,700 --> 00:43:27,720
came in and found also
sophisticated proofs

598
00:43:27,720 --> 00:43:32,230
of Szemeredi's theorem
from other areas

599
00:43:32,230 --> 00:43:34,470
and using other
tools, including--

600
00:43:34,470 --> 00:43:37,140
and here are some of the
most important perspectives,

601
00:43:37,140 --> 00:43:39,180
later perspectives, of
Szemeredi's theorem.

602
00:43:39,180 --> 00:43:44,630
So there was a proof
using ergodic theory

603
00:43:44,630 --> 00:43:48,120
that followed fairly
shortly after Szemeredi's

604
00:43:48,120 --> 00:43:50,810
original proof.

605
00:43:50,810 --> 00:43:52,170
This is due to Furstenberg.

606
00:43:54,780 --> 00:43:57,890
And initially, it wasn't clear,
because all of these proofs

607
00:43:57,890 --> 00:43:59,010
were so involved.

608
00:43:59,010 --> 00:44:03,890
It wasn't clear if the ergodic
theoretic proof was genuinely

609
00:44:03,890 --> 00:44:08,390
something new, or it was a
rephrasing of Szemeredi's

610
00:44:08,390 --> 00:44:10,250
combinatorial proof.

611
00:44:10,250 --> 00:44:12,050
But then very quickly
it was realized

612
00:44:12,050 --> 00:44:15,830
that there were extensions
of Szemeredi's theorem,

613
00:44:15,830 --> 00:44:19,280
other combinatorial results
that the ergodic theorists could

614
00:44:19,280 --> 00:44:23,450
establish using their methods,
so using the same methods

615
00:44:23,450 --> 00:44:25,220
or extensions of
the same methods

616
00:44:25,220 --> 00:44:29,440
that combinatorialists
did not know how to do.

617
00:44:29,440 --> 00:44:31,420
And to this date,
there are still

618
00:44:31,420 --> 00:44:34,690
theorems for which
the only known proofs

619
00:44:34,690 --> 00:44:37,812
use ergodic theory,
so extensions

620
00:44:37,812 --> 00:44:38,770
of Szemeredi's theorem.

621
00:44:38,770 --> 00:44:40,600
And I will mention
one later on today.

622
00:44:44,770 --> 00:44:47,020
So that's one of
the perspectives.

623
00:44:47,020 --> 00:44:50,710
The other perspective that
was also quite influential

624
00:44:50,710 --> 00:44:52,750
there is something known
as higher order Fourier

625
00:44:52,750 --> 00:45:07,000
analysis, which was pioneered
by Tim Gowers' in around 2000.

626
00:45:07,000 --> 00:45:10,840
So Gowers won the
Fields Medal, party

627
00:45:10,840 --> 00:45:12,850
for his work on
Banach spaces but also

628
00:45:12,850 --> 00:45:15,910
party for this development.

629
00:45:15,910 --> 00:45:18,520
So higher order Fourier
analysis is in some sense

630
00:45:18,520 --> 00:45:20,800
an extension of Roth's theorem.

631
00:45:20,800 --> 00:45:23,140
So anyway, Roth also
won a Fields Medal,

632
00:45:23,140 --> 00:45:25,330
although this is not
his most famous term.

633
00:45:25,330 --> 00:45:27,700
I'll say his second
most famous theorem.

634
00:45:27,700 --> 00:45:31,450
So Roth used this
Fourier analysis

635
00:45:31,450 --> 00:45:36,000
in the sense of Hardy-Littlewood
to control 3-term arithmetic

636
00:45:36,000 --> 00:45:37,140
progressions.

637
00:45:37,140 --> 00:45:39,240
But it turns out
that that method

638
00:45:39,240 --> 00:45:41,820
for very good
fundamental reasons

639
00:45:41,820 --> 00:45:45,930
completely fails for 4-term
arithmetic progressions.

640
00:45:45,930 --> 00:45:49,600
So we'll see later in the
course why that's the case,

641
00:45:49,600 --> 00:45:52,080
why is it that you cannot do
Fourier analysis to control

642
00:45:52,080 --> 00:45:53,640
4-term APs.

643
00:45:53,640 --> 00:45:56,970
But Gowers managed to find a
way to overcome that difficulty.

644
00:45:56,970 --> 00:45:58,800
And he came up
with an extension,

645
00:45:58,800 --> 00:46:00,810
with a generalization
of Fourier analysis,

646
00:46:00,810 --> 00:46:03,360
very powerful, very
difficult to use, actually.

647
00:46:03,360 --> 00:46:08,380
But that allows you to
understand longer arithmetic

648
00:46:08,380 --> 00:46:11,570
progressions.

649
00:46:11,570 --> 00:46:15,410
Another very
influential approach

650
00:46:15,410 --> 00:46:17,599
is called hypergraph regularity.

651
00:46:23,090 --> 00:46:26,130
So the hypergraph
regularity method

652
00:46:26,130 --> 00:46:36,190
was also discovered in the early
2000s independently by a team

653
00:46:36,190 --> 00:46:40,830
led by Rodl and also by Gowers.

654
00:46:44,170 --> 00:46:47,220
So the hypergraph
regularity method

655
00:46:47,220 --> 00:46:51,130
is an extension of what's known
as Szemeredi's regularity,

656
00:46:51,130 --> 00:46:53,740
Szemeredi's graph
regularity method.

657
00:46:53,740 --> 00:46:55,360
And this is the
method that will be

658
00:46:55,360 --> 00:47:00,100
a central topic in the
first half of this course.

659
00:47:00,100 --> 00:47:04,120
And it's a method that is
quite central, or at least some

660
00:47:04,120 --> 00:47:07,680
of the ideas quite central,
to Szemeredi's method.

661
00:47:07,680 --> 00:47:09,900
And he gave an
alternative proof.

662
00:47:09,900 --> 00:47:12,970
He and Ruzsa gave an alternative
proof of Roth's theorem

663
00:47:12,970 --> 00:47:15,030
using graph theory.

664
00:47:15,030 --> 00:47:17,210
And for a long time,
people realized

665
00:47:17,210 --> 00:47:20,630
that one could extend some of
those ideas to hypergraphs.

666
00:47:20,630 --> 00:47:23,570
But working out how
that proof goes actually

667
00:47:23,570 --> 00:47:25,730
took an enormous amount
of time and effort

668
00:47:25,730 --> 00:47:31,030
and resulted in this amazing
theorem on hypergraph.

669
00:47:31,030 --> 00:47:34,380
Let me mention these are
not the only methods that

670
00:47:34,380 --> 00:47:36,630
were used to extend
Szemeredi's theorem

671
00:47:36,630 --> 00:47:38,040
or give alternate proofs.

672
00:47:38,040 --> 00:47:39,790
There are many others.

673
00:47:39,790 --> 00:47:41,850
For example, you may
have heard of something

674
00:47:41,850 --> 00:47:44,110
called the polymath project.

675
00:47:44,110 --> 00:47:46,530
Raise your hand if you heard
of the polymath project.

676
00:47:46,530 --> 00:47:47,070
OK, great.

677
00:47:47,070 --> 00:47:49,920
So maybe about half of you.

678
00:47:49,920 --> 00:47:54,030
So this is an online
collaborative project

679
00:47:54,030 --> 00:48:00,150
started by Tim Gowers and also
famous people like Terry Tao.

680
00:48:00,150 --> 00:48:04,170
And they were all quite involved
in various polymath projects.

681
00:48:04,170 --> 00:48:07,800
And the first successful
polymath project

682
00:48:07,800 --> 00:48:11,670
produced a combinatorial
proof of something

683
00:48:11,670 --> 00:48:13,860
known as the density
Hales-Jewett theorem.

684
00:48:20,450 --> 00:48:22,840
So I won't explain
what it this here.

685
00:48:22,840 --> 00:48:26,120
So it's something which
is related to tic tac toe.

686
00:48:26,120 --> 00:48:27,770
But let me not go into that.

687
00:48:27,770 --> 00:48:32,260
So it's a deep combinatorial
theorem that had they

688
00:48:32,260 --> 00:48:34,480
known earlier using
ergodic theoretic methods,

689
00:48:34,480 --> 00:48:36,350
but they gave a new
combinatorial proof,

690
00:48:36,350 --> 00:48:41,200
in particular gave some
concrete bounds on this theorem

691
00:48:41,200 --> 00:48:45,760
and that in particular also
implies Szemeredi's theorem.

692
00:48:45,760 --> 00:48:47,500
So this gave a new proof.

693
00:48:47,500 --> 00:48:50,140
And as a result, they--

694
00:48:50,140 --> 00:48:52,020
it's an online
collaborative project--

695
00:48:52,020 --> 00:48:55,840
so they published this paper
under the pseudonym DHJ

696
00:48:55,840 --> 00:49:03,452
Polymath, where DHJ stands
for Density Hales Jewett.

697
00:49:03,452 --> 00:49:04,910
And they kept the
same name for all

698
00:49:04,910 --> 00:49:06,740
of the subsequent
papers published

699
00:49:06,740 --> 00:49:10,440
by the polymath project.

700
00:49:10,440 --> 00:49:14,180
So as you see through
all of these examples

701
00:49:14,180 --> 00:49:17,750
that there a lot of
work that were motivated

702
00:49:17,750 --> 00:49:19,370
by Szemeredi's theorem.

703
00:49:19,370 --> 00:49:21,460
This is truly a
foundational result,

704
00:49:21,460 --> 00:49:25,190
a foundational
theorem that gave way

705
00:49:25,190 --> 00:49:28,770
to a lot of important research.

706
00:49:28,770 --> 00:49:32,540
And Szemeredi himself
received an Apple Prize

707
00:49:32,540 --> 00:49:36,162
for his seminal contributions
to combinatorics and also

708
00:49:36,162 --> 00:49:37,370
theoretical computer science.

709
00:49:40,460 --> 00:49:42,980
We still don't
understand in some sense

710
00:49:42,980 --> 00:49:44,960
completely what
Szemeredi's theorem--

711
00:49:44,960 --> 00:49:48,410
you know, for example, we do
understand the optimal bounds.

712
00:49:48,410 --> 00:49:50,660
And also more importantly,
conceptually, we

713
00:49:50,660 --> 00:49:53,720
don't really understand
how these methods are

714
00:49:53,720 --> 00:49:55,740
related to each other.

715
00:49:55,740 --> 00:49:57,680
So there's some vague
sense that they all

716
00:49:57,680 --> 00:50:00,100
have some common things.

717
00:50:00,100 --> 00:50:06,340
But there is a lot of mystery as
to what do these methods coming

718
00:50:06,340 --> 00:50:08,460
from very different areas--

719
00:50:08,460 --> 00:50:10,278
ergodic theory,
harmonic analysis, you

720
00:50:10,278 --> 00:50:12,070
what do they all have
to do with each other

721
00:50:12,070 --> 00:50:14,260
but there is central theme.

722
00:50:14,260 --> 00:50:20,110
And this is also going to be
a theme in this course, which

723
00:50:20,110 --> 00:50:21,220
goes under the name--

724
00:50:21,220 --> 00:50:24,520
and I believe Terry Tao is the
one who popularized this name--

725
00:50:24,520 --> 00:50:35,420
the dichotomy between structure
and randomness, structure

726
00:50:35,420 --> 00:50:37,387
and pseudo randomness.

727
00:50:42,060 --> 00:50:46,930
Somehow it's a really fancy way
of saying signal versus noise.

728
00:50:46,930 --> 00:50:51,640
So I give you some object, I
give you some complex object,

729
00:50:51,640 --> 00:50:53,740
and there is some
mathematical way

730
00:50:53,740 --> 00:50:59,980
to separate the structure
from some noisy aspects, which

731
00:50:59,980 --> 00:51:01,690
behave random-like.

732
00:51:01,690 --> 00:51:03,760
So there will be many
places in this course

733
00:51:03,760 --> 00:51:09,210
where this dichotomy will
play an important role.

734
00:51:09,210 --> 00:51:11,010
Any questions at this point?

735
00:51:20,030 --> 00:51:24,020
I want to take you through some
generalizations and extensions

736
00:51:24,020 --> 00:51:26,630
of Szemeredi's theorem.

737
00:51:26,630 --> 00:51:28,940
So first, let's
look at what happens

738
00:51:28,940 --> 00:51:31,100
if we go to higher dimensions.

739
00:51:34,970 --> 00:51:39,100
Suppose we have a
subset in D dimensions,

740
00:51:39,100 --> 00:51:41,190
d-dimensional lattice.

741
00:51:41,190 --> 00:51:44,250
So we can also define
some notion of density.

742
00:51:44,250 --> 00:51:46,200
Again, it doesn't
matter precisely what

743
00:51:46,200 --> 00:51:47,680
is the notion you use.

744
00:51:47,680 --> 00:51:57,520
For example, we can say that
it has a positive upper density

745
00:51:57,520 --> 00:52:12,882
if this lim sup is positive.

746
00:52:16,260 --> 00:52:19,460
So Szemeredi's theorem
in one dimension

747
00:52:19,460 --> 00:52:22,335
tells us that if you have some
sort of positive density, then

748
00:52:22,335 --> 00:52:26,550
I can find arbitrarily long
arithmetic progressions.

749
00:52:26,550 --> 00:52:28,860
So what should the
corresponding generalization

750
00:52:28,860 --> 00:52:31,760
in higher dimensions?

751
00:52:31,760 --> 00:52:34,350
Well, here's a notion
that I can define, namely

752
00:52:34,350 --> 00:52:55,030
that we say that a contains
arbitrary constellations

753
00:52:55,030 --> 00:52:57,972
to mean that--

754
00:52:57,972 --> 00:52:58,930
so what does that mean?

755
00:52:58,930 --> 00:53:00,555
So a constellation,
you can think of it

756
00:53:00,555 --> 00:53:04,920
as some finite pattern, so a
set of stars in the sky, so

757
00:53:04,920 --> 00:53:05,500
some pattern.

758
00:53:05,500 --> 00:53:09,970
And I want to find that
pattern somewhere in a, where

759
00:53:09,970 --> 00:53:11,360
I'm allowed to dilate.

760
00:53:11,360 --> 00:53:16,970
So I'm allowed to do to
multiply pattern by some number

761
00:53:16,970 --> 00:53:18,206
and also translate.

762
00:53:18,206 --> 00:53:20,320
So on the finite pattern--

763
00:53:20,320 --> 00:53:27,150
so what I mean precisely is
that for every finite subset

764
00:53:27,150 --> 00:53:40,420
of the grid, there exists some
translation and some dilation,

765
00:53:40,420 --> 00:53:47,200
such that once I apply this
dilation and translation

766
00:53:47,200 --> 00:53:56,490
to my pattern F, meaning I'm
looking at the image of this F

767
00:53:56,490 --> 00:54:00,730
under this transformation,
then this set lies inside a.

768
00:54:03,901 --> 00:54:07,030
So you see that
arithmetic progressions

769
00:54:07,030 --> 00:54:11,040
is the constellation,
just numbers 1 through k.

770
00:54:14,020 --> 00:54:15,822
So that's a definition.

771
00:54:15,822 --> 00:54:17,780
And the multi-dimensional
Szemeredi's theorem--

772
00:54:26,040 --> 00:54:27,910
so the multi-dimensional
generalization

773
00:54:27,910 --> 00:54:37,500
of Szemeredi's theorem says
that for every subset--

774
00:54:37,500 --> 00:54:43,980
so every subset of the
d-dimensional lattice

775
00:54:43,980 --> 00:54:54,350
of possible density contains
arbitrary constellations.

776
00:55:04,170 --> 00:55:06,540
You give me a pattern,
and I can find

777
00:55:06,540 --> 00:55:11,180
this pattern inside a, provided
that a has positive density.

778
00:55:14,350 --> 00:55:21,010
So in particular, if I want to
find a 10 by 10 square grid,

779
00:55:21,010 --> 00:55:26,910
so meaning suppose I want to
find a pattern which consists

780
00:55:26,910 --> 00:55:35,380
of something like that, a
10 by 10 square grid, where

781
00:55:35,380 --> 00:55:38,710
all of these lengths
are equal, but I

782
00:55:38,710 --> 00:55:40,000
don't specify what they are.

783
00:55:40,000 --> 00:55:43,450
But as long as they are equal,
then the theorem tells me

784
00:55:43,450 --> 00:55:46,810
that as long as a
has positive density,

785
00:55:46,810 --> 00:55:51,100
then I can find such
a pattern inside a.

786
00:55:51,100 --> 00:55:55,300
So this theorem was proved
by Furstenberg and Ketsen.

787
00:55:59,860 --> 00:56:03,530
So you see that it is a
generalization of Szemeredi's.

788
00:56:03,530 --> 00:56:06,530
So the one-dimensional case is
precisely Szemeredi's theorem.

789
00:56:10,490 --> 00:56:13,580
So Furstenberg and Ketsen,
using ergodic theory

790
00:56:13,580 --> 00:56:18,510
showed that one can
generalize Szemeredi's theorem

791
00:56:18,510 --> 00:56:20,740
to the multi-dimensional
setting.

792
00:56:20,740 --> 00:56:23,610
However, the
combinatorial approaches

793
00:56:23,610 --> 00:56:26,680
employed by Szemeredi did
not easily generalize.

794
00:56:26,680 --> 00:56:30,600
So it took another couple
of decades at least

795
00:56:30,600 --> 00:56:33,840
for people to find a
combinatorial proof

796
00:56:33,840 --> 00:56:36,780
of this result. And
namely that happened

797
00:56:36,780 --> 00:56:40,750
with the hypergraph
regularity method.

798
00:56:40,750 --> 00:56:44,305
So this was one of the
motivations of this project.

799
00:56:44,305 --> 00:56:45,680
And you say, OK,
what's the point

800
00:56:45,680 --> 00:56:46,805
of having different proofs?

801
00:56:46,805 --> 00:56:50,540
Well, for one thing it's nice
to know different perspectives

802
00:56:50,540 --> 00:56:52,820
to important theorem.

803
00:56:52,820 --> 00:56:56,610
But there's also
concrete objective.

804
00:56:56,610 --> 00:57:00,150
In particular, it turns out
that if you prove something

805
00:57:00,150 --> 00:57:04,100
using ergodic theory, because--

806
00:57:04,100 --> 00:57:06,690
we will not discuss ergodic
theory in this course.

807
00:57:06,690 --> 00:57:10,550
But roughly, one of the
early steps in such a proof

808
00:57:10,550 --> 00:57:13,330
applies compactness.

809
00:57:13,330 --> 00:57:16,150
And that already
destroys any chance

810
00:57:16,150 --> 00:57:18,940
of getting concrete
quantitative bounds.

811
00:57:18,940 --> 00:57:23,920
So you can ask if I want
to find a 10 by 10 pattern

812
00:57:23,920 --> 00:57:32,090
and I have density 1%, how
large do I need to look?

813
00:57:32,090 --> 00:57:34,980
How far do I have to look in
order to find that pattern?

814
00:57:34,980 --> 00:57:36,710
So that's a
quantitative question

815
00:57:36,710 --> 00:57:39,790
that is actually not at all
addressed by ergodic theory.

816
00:57:39,790 --> 00:57:44,500
So the later methods using
combinatorial methods

817
00:57:44,500 --> 00:57:46,410
gave you concrete bounds.

818
00:57:46,410 --> 00:57:49,310
And so there are some
concrete differences

819
00:57:49,310 --> 00:57:50,380
between these methods.

820
00:57:53,930 --> 00:57:57,040
So this theorem
reminds me of the scene

821
00:57:57,040 --> 00:57:59,400
from the movie a
Beautiful Mind, which

822
00:57:59,400 --> 00:58:02,700
is one of the greatest
mathematical movies

823
00:58:02,700 --> 00:58:04,450
in some sense.

824
00:58:04,450 --> 00:58:08,020
And so there's a scene
there where Russell Crowe

825
00:58:08,020 --> 00:58:11,640
playing John Nash--

826
00:58:11,640 --> 00:58:13,730
so there were at
this fancy party.

827
00:58:13,730 --> 00:58:17,780
And Nash was with his
soon to be wife, Alicia.

828
00:58:17,780 --> 00:58:21,670
And he points to the sky
and tells her, pick a shape.

829
00:58:21,670 --> 00:58:24,683
Pick a shape and I can find
for you among the stars.

830
00:58:24,683 --> 00:58:26,850
And so this is what the
theorem allows you to do it.

831
00:58:31,340 --> 00:58:33,540
So give me a shape
and I can find

832
00:58:33,540 --> 00:58:35,230
that constellation inside a.

833
00:58:38,720 --> 00:58:41,464
Let's look at other
generalizations.

834
00:58:44,430 --> 00:58:47,120
So far, we are looking
at linear patterns.

835
00:58:47,120 --> 00:58:50,070
So we're looking at linear
dilations and translations.

836
00:58:50,070 --> 00:58:53,040
But what about
polynomial patterns?

837
00:58:53,040 --> 00:58:54,450
So here's a question.

838
00:58:54,450 --> 00:58:58,230
Suppose I give you a dense
subset, a positive density

839
00:58:58,230 --> 00:59:00,240
subset of integers.

840
00:59:00,240 --> 00:59:06,590
Can you find two numbers whose
difference is a perfect square?

841
00:59:06,590 --> 00:59:08,470
So this question
was asked by Lovasz.

842
00:59:08,470 --> 00:59:14,310
And a positive answer was given
in the late '70s by Furstenberg

843
00:59:14,310 --> 00:59:16,749
and Sarkozy independently.

844
00:59:28,485 --> 00:59:33,460
So Furstenberg and Sarkozy,
they showed using different

845
00:59:33,460 --> 00:59:37,280
methods-- so one ergodic
theoretic and the other is more

846
00:59:37,280 --> 00:59:39,340
harmonic analytic--

847
00:59:39,340 --> 00:59:47,110
that every subset of the
integers, so every subset

848
00:59:47,110 --> 00:59:56,460
of positive integers,
with positive density

849
00:59:56,460 --> 01:00:04,530
contains two numbers
differing by a perfect square.

850
01:00:14,120 --> 01:00:15,800
So in other words,
we can always find

851
01:00:15,800 --> 01:00:19,340
the pattern x plus y squared.

852
01:00:22,810 --> 01:00:24,670
So what about other
polynomial patterns?

853
01:00:24,670 --> 01:00:27,130
Instead of this y
squared, suppose you just

854
01:00:27,130 --> 01:00:30,820
give me some other
polynomial or maybe

855
01:00:30,820 --> 01:00:33,500
a collection of polynomials.

856
01:00:33,500 --> 01:00:34,960
So what can I say?

857
01:00:34,960 --> 01:00:38,110
Well, there are some things
for which this is not true.

858
01:00:38,110 --> 01:00:40,210
Can you give me an
example where if I

859
01:00:40,210 --> 01:00:42,852
putting the wrong
polynomial it's not true?

860
01:00:46,120 --> 01:00:48,100
What if the polynomial
is the constant 1?

861
01:00:52,020 --> 01:00:55,880
If you take the even
numbers, has density 1/2,

862
01:00:55,880 --> 01:00:59,520
but it doesn't contain any
patterns of x and x plus 1.

863
01:00:59,520 --> 01:01:03,610
So I need to say some hypotheses
about these polynomials.

864
01:01:03,610 --> 01:01:07,200
So a vast generalization
of this result,

865
01:01:07,200 --> 01:01:17,420
so known as polynomial
Szemeredi theorem,

866
01:01:17,420 --> 01:01:30,340
says that if A is a subset of
integers with positive density,

867
01:01:30,340 --> 01:01:38,860
and if we have these
polynomials, P1 through Pk

868
01:01:38,860 --> 01:01:51,460
with integer coefficients
and zero constant terms,

869
01:01:51,460 --> 01:01:54,520
then I can always
find a pattern.

870
01:01:54,520 --> 01:02:02,200
So there exists some x
and positive integer y

871
01:02:02,200 --> 01:02:10,140
such that this pattern, x
plus P1 of y, x plus P2 of y,

872
01:02:10,140 --> 01:02:18,530
and so on x, plus Pk of
y, they all lie in A.

873
01:02:18,530 --> 01:02:22,630
So in other words, succinctly,
every subset of integers

874
01:02:22,630 --> 01:02:25,885
with positive density contains
arbitrary polynomial patterns.

875
01:02:29,240 --> 01:02:30,628
So this was proved--

876
01:02:30,628 --> 01:02:32,170
so this was an
important result proof

877
01:02:32,170 --> 01:02:41,090
by Bergelson and Liebman
using ergodic theory.

878
01:02:41,090 --> 01:02:44,340
And so far for this
general statement,

879
01:02:44,340 --> 01:02:49,382
the only known proof
uses ergodic theory.

880
01:02:49,382 --> 01:02:50,965
So there was some
recent developments,

881
01:02:50,965 --> 01:02:52,423
recent pretty
exciting developments

882
01:02:52,423 --> 01:02:55,280
that for some
specific cases where

883
01:02:55,280 --> 01:02:58,460
if you have some additional
restrictions on the P's, then

884
01:02:58,460 --> 01:03:01,130
there are other methods
coming from Fourier analytic,

885
01:03:01,130 --> 01:03:04,372
harmonic analytic methods
that could give you

886
01:03:04,372 --> 01:03:06,580
a different proof that allows
you to get some bounds.

887
01:03:06,580 --> 01:03:11,260
Remember, the ergodic
proof gives you no bounds.

888
01:03:11,260 --> 01:03:14,150
But so far, in general,
the only method known

889
01:03:14,150 --> 01:03:16,000
is ergodic theoretic.

890
01:03:16,000 --> 01:03:18,110
And actually,
Bergelson and Liebman

891
01:03:18,110 --> 01:03:21,080
proved something which is more
general than what I've stated.

892
01:03:21,080 --> 01:03:26,278
So this is also true in a
multidimensional setting.

893
01:03:26,278 --> 01:03:28,820
I won't state that precisely,
but you can imagine what it is.

894
01:03:33,270 --> 01:03:39,347
Let me mention one more theorem
that many of you I imagine

895
01:03:39,347 --> 01:03:39,930
have heard of.

896
01:03:42,343 --> 01:03:43,760
And this is the
Green Tao theorem.

897
01:03:48,853 --> 01:03:54,390
So the Green Tao theorem
says that the primes

898
01:03:54,390 --> 01:03:57,525
contain arbitrarily long
arithmetic progressions.

899
01:04:07,440 --> 01:04:09,210
So this is a famous theorem.

900
01:04:09,210 --> 01:04:12,840
And it's one of the
most celebrated results

901
01:04:12,840 --> 01:04:14,460
of the past couple of decades.

902
01:04:14,460 --> 01:04:16,660
And it resolved some
longstanding folklore

903
01:04:16,660 --> 01:04:19,350
conjectures in number theory.

904
01:04:19,350 --> 01:04:21,790
The Green Tao
theorem, well, you see

905
01:04:21,790 --> 01:04:25,480
that in form it looks somewhat
like Szemeredi 's theorem.

906
01:04:25,480 --> 01:04:28,110
But it doesn't follow
from Szemeredi 's theorem.

907
01:04:28,110 --> 01:04:30,555
Well, the primes, they
don't have positive density.

908
01:04:30,555 --> 01:04:31,930
The prime number
theorem tells us

909
01:04:31,930 --> 01:04:35,870
that density decays
like 1 over log n.

910
01:04:35,870 --> 01:04:39,200
So what about quantity versions
of Szemeredi 's theorem?

911
01:04:39,200 --> 01:04:40,910
It is possible.

912
01:04:40,910 --> 01:04:43,580
Although we do not know how
to prove such statement,

913
01:04:43,580 --> 01:04:46,370
it is possible that
a density of primes

914
01:04:46,370 --> 01:04:51,930
alone might guarantee the
Green Tao theorem in that it

915
01:04:51,930 --> 01:04:54,030
is possible that
Szemeredi 's theorem

916
01:04:54,030 --> 01:04:59,010
is true for any set
whose density decays

917
01:04:59,010 --> 01:05:02,100
like the prime numbers,
like 1 over log n.

918
01:05:02,100 --> 01:05:04,890
But no we're quite far from
proving such a statement.

919
01:05:04,890 --> 01:05:08,330
And that's not what
Green and Tao did.

920
01:05:08,330 --> 01:05:14,560
Instead, they took Szemeredi
's theorem as a black box

921
01:05:14,560 --> 01:05:19,520
and applied it to some
variant of the primes

922
01:05:19,520 --> 01:05:23,240
and showed that
inside this variant,

923
01:05:23,240 --> 01:05:25,070
Szemeredi 's theorem
is also true,

924
01:05:25,070 --> 01:05:28,890
and that the primes sit inside
this variant of the primes,

925
01:05:28,890 --> 01:05:32,880
known as pseudo primes, as a set
of relatively positive density,

926
01:05:32,880 --> 01:05:35,060
but somehow
transferring Szemeredi

927
01:05:35,060 --> 01:05:38,940
's theorem from the dense
setting to a sparser setting.

928
01:05:38,940 --> 01:05:42,110
So this is a very
exciting technique.

929
01:05:42,110 --> 01:05:46,160
And as a result, Green-Tao
proved not just that the primes

930
01:05:46,160 --> 01:05:49,340
contain arbitrarily long
arithmetic progressions,

931
01:05:49,340 --> 01:05:56,480
but every relatively dense,
so relatively positive

932
01:05:56,480 --> 01:06:05,120
density subset, of the
primes contains arbitrarily

933
01:06:05,120 --> 01:06:07,053
long arithmetic progressions.

934
01:06:11,170 --> 01:06:13,240
To prove this theorem
they incorporated

935
01:06:13,240 --> 01:06:17,850
many different ideas coming
from many different areas

936
01:06:17,850 --> 01:06:20,790
of mathematics, including
harmonic analysis,

937
01:06:20,790 --> 01:06:24,870
some ideas coming from
combinatorics, and number

938
01:06:24,870 --> 01:06:25,690
theory as well.

939
01:06:25,690 --> 01:06:28,710
So there were some innovations
at the time in number theory

940
01:06:28,710 --> 01:06:31,180
that were employed
in this result.

941
01:06:31,180 --> 01:06:34,350
So this is certainly
a landmark theorem.

942
01:06:34,350 --> 01:06:38,520
And although we will
not discuss a full proof

943
01:06:38,520 --> 01:06:41,680
of the Green-Tao theorem, we
will go into some of the ideas

944
01:06:41,680 --> 01:06:42,780
through this course.

945
01:06:42,780 --> 01:06:44,725
And I will show
you bits and pieces

946
01:06:44,725 --> 01:06:46,350
that we will see
throughout the course.

947
01:06:48,980 --> 01:06:53,740
So this is meant to be
a very fast tour of what

948
01:06:53,740 --> 01:06:57,550
happened in the last 100 years
in additive combinatorics,

949
01:06:57,550 --> 01:07:00,280
taking you from Schur's
theorem, which was really

950
01:07:00,280 --> 01:07:02,680
about 100 years
ago, to something

951
01:07:02,680 --> 01:07:05,710
that is much more modern.

952
01:07:05,710 --> 01:07:08,680
But now, instead of
being up in the stars,

953
01:07:08,680 --> 01:07:11,250
let's come back down to Earth.

954
01:07:11,250 --> 01:07:13,940
And I want to talk about
what we'll do next.

955
01:07:13,940 --> 01:07:16,720
So what are some of the
things that we can actually

956
01:07:16,720 --> 01:07:23,200
prove that doesn't
involve taking up 50 pages

957
01:07:23,200 --> 01:07:27,070
using a complex logical diagram,
as Szemeredi did in his paper.

958
01:07:27,070 --> 01:07:29,700
So what are some of the simple
things that we can start with?

959
01:07:29,700 --> 01:07:33,440
Well, so first, let's go
back to Roth's theorem.

960
01:07:33,440 --> 01:07:36,670
So Roth's theorem, we
stated it up there.

961
01:07:36,670 --> 01:07:39,385
But let me restate it
in a finitary form.

962
01:07:43,256 --> 01:07:45,580
So Roth's theorem
is the statement

963
01:07:45,580 --> 01:07:52,960
that every subset of
integers 1 through n that

964
01:07:52,960 --> 01:07:58,630
avoids 3-term
arithmetic progressions

965
01:07:58,630 --> 01:08:05,380
must have size little
o of N. So earlier we

966
01:08:05,380 --> 01:08:08,110
gave an infinitary
statement that if you

967
01:08:08,110 --> 01:08:10,470
have a positive density
subset of the integers

968
01:08:10,470 --> 01:08:12,340
that it contains
a three AP, this

969
01:08:12,340 --> 01:08:16,359
is an equivalent
finitary statement.

970
01:08:16,359 --> 01:08:19,930
Roth's original proof
used Fourier analysis.

971
01:08:19,930 --> 01:08:24,220
And a different proof was
given in the '70s by Rusza

972
01:08:24,220 --> 01:08:27,460
and Szemeredi using
graph theoretic methods.

973
01:08:27,460 --> 01:08:31,270
So how does graph theory
have to do with this result?

974
01:08:31,270 --> 01:08:34,060
And this shouldn't be
surprising to this point, given

975
01:08:34,060 --> 01:08:38,830
that we already saw how we
used Ramsay's theorem, graph

976
01:08:38,830 --> 01:08:40,750
theoretic result, to
prove Schur's theorem,

977
01:08:40,750 --> 01:08:42,800
which is something that
is number theoretic.

978
01:08:42,800 --> 01:08:46,029
So something similar happens.

979
01:08:46,029 --> 01:08:50,120
But now, the question is what
is the graph theoretic problem

980
01:08:50,120 --> 01:08:52,029
that we need to look at?

981
01:08:52,029 --> 01:08:56,000
So for Schur's theorem it was
Ramsey's theorem for triangles.

982
01:08:56,000 --> 01:08:58,750
But what about for
Roth's theorem?

983
01:08:58,750 --> 01:09:02,930
A naive guess is the following.

984
01:09:02,930 --> 01:09:05,050
So what's the question
that we should ask?

985
01:09:07,700 --> 01:09:11,930
Here's a somewhat naive
guess, which turns out

986
01:09:11,930 --> 01:09:13,670
not to be the right
question, but still

987
01:09:13,670 --> 01:09:15,649
an interesting
question, which is

988
01:09:15,649 --> 01:09:30,662
what is the maximum number of
edges in a triangle-free graph

989
01:09:30,662 --> 01:09:31,659
on n vertices?

990
01:09:37,149 --> 01:09:39,109
Now, this is not
totally a stupid guess,

991
01:09:39,109 --> 01:09:44,569
because as you imagine from what
we said with Schur's theorem,

992
01:09:44,569 --> 01:09:47,600
somehow you want to
set up a graph so

993
01:09:47,600 --> 01:09:50,760
that the triangles correspond
to the 3-term arithmetic

994
01:09:50,760 --> 01:09:52,100
progressions.

995
01:09:52,100 --> 01:09:53,930
And you want to set
it up in such a way

996
01:09:53,930 --> 01:09:57,890
that this question about what's
the maximum size subset of 1

997
01:09:57,890 --> 01:10:00,350
through n without
3 APs translates

998
01:10:00,350 --> 01:10:03,350
into some question about what's
the maximum number of edges

999
01:10:03,350 --> 01:10:07,155
in a graph that
has some property?

1000
01:10:07,155 --> 01:10:09,220
So what is that property?

1001
01:10:09,220 --> 01:10:12,120
So this is not a
totally stupid guess.

1002
01:10:12,120 --> 01:10:16,320
But it turns out this
question is relatively easy.

1003
01:10:16,320 --> 01:10:17,540
Still it has a name.

1004
01:10:17,540 --> 01:10:23,570
So this was found by
Mantel about 100 years ago,

1005
01:10:23,570 --> 01:10:24,790
so known as Mantel's theorem.

1006
01:10:28,350 --> 01:10:32,432
And the answer, well,
we'll see a proof.

1007
01:10:32,432 --> 01:10:34,390
So the first thing we'll
do in the next lecture

1008
01:10:34,390 --> 01:10:39,210
is prove Mantel's theorem, but
I do want to hold suspense.

1009
01:10:39,210 --> 01:10:40,900
I mean the answer,
it turns out to be

1010
01:10:40,900 --> 01:10:42,490
fairly simple to describe.

1011
01:10:42,490 --> 01:10:45,130
Namely that you split
the vertices into two

1012
01:10:45,130 --> 01:10:46,930
basically equal halves.

1013
01:10:46,930 --> 01:10:52,900
And you join all the possible
edges between the two halves.

1014
01:10:52,900 --> 01:10:58,290
So this complete bipartide graph
with two equal-sized parts.

1015
01:10:58,290 --> 01:11:01,438
And it turns out this graph,
you see this triangle-free

1016
01:11:01,438 --> 01:11:03,730
and also turns out to have
the maximum number of edges.

1017
01:11:03,730 --> 01:11:04,945
Yeah, question.

1018
01:11:04,945 --> 01:11:07,735
AUDIENCE: What are asymptotics
for three arithmetic

1019
01:11:07,735 --> 01:11:08,458
progression of--

1020
01:11:08,458 --> 01:11:10,250
YUFEI ZHAO: Let me get
to that in a second.

1021
01:11:10,250 --> 01:11:14,186
OK, so I'll talk about
asymptotics in a second.

1022
01:11:14,186 --> 01:11:16,910
So it turns that this is not the
right graph theoretic question

1023
01:11:16,910 --> 01:11:17,410
to ask.

1024
01:11:17,410 --> 01:11:21,298
So what is the right graph
theoretic question to ask?

1025
01:11:21,298 --> 01:11:22,340
I'll tell you what it is.

1026
01:11:22,340 --> 01:11:25,210
I mean it shouldn't be
clear to you at this point.

1027
01:11:25,210 --> 01:11:28,450
It still seems like an
interesting question,

1028
01:11:28,450 --> 01:11:32,720
but it's also somewhat bizarre
to think about if you've never

1029
01:11:32,720 --> 01:11:34,170
seen this before.

1030
01:11:34,170 --> 01:11:39,750
So what is the maximum
number of edges

1031
01:11:39,750 --> 01:11:53,290
in an n vertex graph, where
every edge lies in exactly one

1032
01:11:53,290 --> 01:11:53,790
triangle?

1033
01:12:03,270 --> 01:12:06,050
So I want a graph with
lots and lots of edges

1034
01:12:06,050 --> 01:12:10,140
where every edge sits
in exactly one triangle.

1035
01:12:10,140 --> 01:12:12,530
Now, you might have
some difficulty even

1036
01:12:12,530 --> 01:12:15,110
coming up with good graphs
that have this property.

1037
01:12:15,110 --> 01:12:15,890
And that's OK.

1038
01:12:15,890 --> 01:12:19,290
These are very strange
things to think about.

1039
01:12:19,290 --> 01:12:21,820
But we'll see many
examples of it later on.

1040
01:12:21,820 --> 01:12:27,120
We'll also see
how Roth's theorem

1041
01:12:27,120 --> 01:12:30,730
is connected to this
graph theoretic question.

1042
01:12:30,730 --> 01:12:32,500
Just to give you
a hint, you know,

1043
01:12:32,500 --> 01:12:35,050
where does exactly one
triangle come from,

1044
01:12:35,050 --> 01:12:39,370
it's because even if you avoid
3-term arithmetic progressions,

1045
01:12:39,370 --> 01:12:41,770
there are still these
trivial 3-term arithmetic

1046
01:12:41,770 --> 01:12:45,910
progressions, where you keep
the same number three times.

1047
01:12:45,910 --> 01:12:47,330
And in graph
theoretic world, that

1048
01:12:47,330 --> 01:12:51,961
comes to the unique triangle
that every edge sits on.

1049
01:12:55,120 --> 01:13:01,450
So to address the question
about quantitative bounds,

1050
01:13:01,450 --> 01:13:06,080
for Roth's theorem, it
turns out that we have

1051
01:13:06,080 --> 01:13:07,880
upper bounds and lower bounds.

1052
01:13:07,880 --> 01:13:10,160
And it is still a
wide open question

1053
01:13:10,160 --> 01:13:13,310
as to what these
things should be.

1054
01:13:13,310 --> 01:13:16,820
And roughly speaking,
the best lower bound

1055
01:13:16,820 --> 01:13:18,860
comes from a construction,
which we'll see later

1056
01:13:18,860 --> 01:13:25,810
in this course, the higher size
around n divided by e to the c

1057
01:13:25,810 --> 01:13:28,210
root log n.

1058
01:13:28,210 --> 01:13:40,745
And the best upper bound is of
the form roughly n over log n.

1059
01:13:40,745 --> 01:13:42,370
That's maybe a little
bit hard to think

1060
01:13:42,370 --> 01:13:44,480
about how these numbers behave.

1061
01:13:44,480 --> 01:13:47,590
So if you raise both
sides to-- the denominator

1062
01:13:47,590 --> 01:13:50,970
to e to the something, then
it's maybe easier to compare.

1063
01:13:50,970 --> 01:13:52,320
But it's still a pretty far gap.

1064
01:13:52,320 --> 01:13:53,470
So still a pretty big gap.

1065
01:13:56,360 --> 01:13:58,590
There's a famous conjecture
of Erdos some of you

1066
01:13:58,590 --> 01:14:00,430
might have heard
of, that if you have

1067
01:14:00,430 --> 01:14:02,290
a subset of the
positive integers

1068
01:14:02,290 --> 01:14:05,440
with divergent
harmonic series, then

1069
01:14:05,440 --> 01:14:10,090
it contains arbitrarily
long automatic progressions.

1070
01:14:10,090 --> 01:14:11,830
That's a very
attractive statement.

1071
01:14:11,830 --> 01:14:14,440
But somehow I don't like
the statement so much,

1072
01:14:14,440 --> 01:14:17,140
because it seems to
make a too pretty.

1073
01:14:17,140 --> 01:14:18,940
And the statement
really is about what

1074
01:14:18,940 --> 01:14:24,430
is the bounds on Roth's theorem
and on Szemeredi 's theorem.

1075
01:14:24,430 --> 01:14:27,610
And having divergent
harmonic series

1076
01:14:27,610 --> 01:14:34,030
is roughly the same as trying
to prove Roth's theorem slightly

1077
01:14:34,030 --> 01:14:36,490
better than the bound
that we currently have,

1078
01:14:36,490 --> 01:14:38,500
somehow breaking this
logarithmic barrier.

1079
01:14:38,500 --> 01:14:41,950
So that conjecture, that having
divergent harmonic series,

1080
01:14:41,950 --> 01:14:45,548
implies 3-term
APs is still open.

1081
01:14:45,548 --> 01:14:46,340
That is still open.

1082
01:14:46,340 --> 01:14:48,940
So where the bound's very
close to what we can prove,

1083
01:14:48,940 --> 01:14:51,160
but it is still open.

1084
01:14:51,160 --> 01:14:54,710
For this question, we will
see later in this course,

1085
01:14:54,710 --> 01:14:59,380
once we've developed Szemeredi
's regularity lemma that we

1086
01:14:59,380 --> 01:15:07,310
can prove an upper bound of o
to the n squared, so little n.

1087
01:15:07,310 --> 01:15:11,916
And that will suffice for
proving Roth's theorem.

1088
01:15:11,916 --> 01:15:13,450
It turns out that
we don't know what

1089
01:15:13,450 --> 01:15:14,680
the right answer should be.

1090
01:15:14,680 --> 01:15:18,180
We don't know what is
the best such graph.

1091
01:15:18,180 --> 01:15:20,790
And it turns out the
best construction

1092
01:15:20,790 --> 01:15:25,590
for this graph there comes from
over here, the best lower bound

1093
01:15:25,590 --> 01:15:28,710
construction of a
set, of a large set

1094
01:15:28,710 --> 01:15:31,090
without 3-term
arithmetic progressions.

1095
01:15:31,090 --> 01:15:34,350
So I'm giving you
a preview of more

1096
01:15:34,350 --> 01:15:37,050
of these connections between
additive combinatorics on one

1097
01:15:37,050 --> 01:15:40,370
hand and graph theory
on the other hand

1098
01:15:40,370 --> 01:15:42,750
that we'll see
throughout this course.

1099
01:15:45,690 --> 01:15:48,020
Any questions?

1100
01:15:48,020 --> 01:15:48,560
OK.

1101
01:15:48,560 --> 01:15:50,610
So just to tell you what's
going to happen next,

1102
01:15:50,610 --> 01:15:52,820
so the next thing that
we're going to discuss

1103
01:15:52,820 --> 01:15:55,480
is basically extremal
graph theory.

1104
01:15:55,480 --> 01:15:59,360
And in particular, if you
forbid some structure,

1105
01:15:59,360 --> 01:16:03,900
such as a triangle, maybe a four
cycle, maybe some other graph,

1106
01:16:03,900 --> 01:16:07,020
what can you say about the
maximum number of edges?

1107
01:16:07,020 --> 01:16:11,207
And there are still a lot of
interesting open problems,

1108
01:16:11,207 --> 01:16:11,790
even for that.

1109
01:16:11,790 --> 01:16:15,940
I forbid some H. What's the
maximum number of edges?

1110
01:16:15,940 --> 01:16:20,360
So the next few lectures
will be on that topic.