YUFEI ZHAO: OK,
let's, get started.
Welcome to 18.217.
So this is combinatorial
theory, graph theory,
and additive combinatorics.
So course website is up there.
So all the course
information is on there.
So after around the
middle of the class,
I'll say a bit more about
various course information,
administrative things.
But I want to jump directly
into the mathematical content.
So this course
roughly has two parts.
The first part will
look at graph theory,
in particular problems
in extremal graph theory.
In the second part,
we'll transition
to additive combinatorics.
But these are not two
separate subjects.
So I want to show you
this topic in a way that
connects these two
areas and show you
that they are quite
related to each other.
And many of the
common themes that
will come up in one
part of the course
will also show up in the other.
So the story between
graph theory and additive
combinatorics began
about 100 years ago
with Schur, the famous
mathematician, Isaai Schur.
Well, he was like
many mathematicians
of his era trying to prove
Fermat's Last Theorem.
So here's what's
Schur's approach.
He said, well, let's look at
this equation that comes up
in from Fermat's Last Theorem.
And, well, one of the methods
of elementary number theory
to rule out solutions
to an equation
is to consider what
happens when you mod p.
If you can rule out for
infinitely many values
p, possible non-trivial
solutions to this equation mod
p, then you will rule out
possibilities of solutions
to Fermat's Last Theorem.
OK, so this was
Schur's approach.
As you can guess, unfortunately,
this approach did not work.
And Schur proved that this
method definitely doesn't work.
So that's the starting
point of our discussion.
So it turns out that
for every value of n,
there exists
non-trivial solutions
for all p sufficiently large.
So thereby, ruling
out the strategy.
So let's see how Schur
proved his theorem.
So that will be the first
half of today's lecture.
So this seems like a
number theory question.
So what does it have to
do with graph theory?
So I wanted to show
you this connection.
Now, Schur deduced his
theorem from another result.
That is known as
Schur's Theorem, which
says that if be
positive integers
is colored using
finitely many colors,
then there exists a
monochromatic solution
to the equation x
plus y equals to z.
So if you give me
10 colors and color
the positive integers
using those 10 colors,
then I can find
for you a solution
to this equation where x, y,
and z are all of the same color.
Now, this statement--
OK, so it's a perfectly
understandable statement.
But let me rephrase it in
a somewhat different way.
And this gets to a
point that I want
to discuss where many statements
in additive combinatorics
or just combinatorics in general
have different formulations,
one that comes in an
infinitary form, which
is more qualitative so to
speak and another form that
is known as finitary.
And that's more
quantitative in nature.
So Schur's Theorem is
stated in a infinitary form.
So it tells you if you color
using finitely many colors,
then there exists a
monochromatic solution.
So many, but not all,
statements of that form
have an equivalent finitary form
that is sometimes more useful.
And also, once you stay
the right finitary form,
you can ask
additional questions.
So here's what Schur's Theorem
looks like in the equivalent
finitary form.
You give me an r.
For every r, there exists
some N as a function of r,
such that if the
numbers 1 through N--
so throughout this course, I'm
going to use this bracket N
to denote integers up to N--
so if these numbers are
colored using our colors,
then necessarily, there exists
a monochromatic solution
to the equation x
plus y equals to z,
where x, y, and z are in the
set that is being colored.
So it looks very similar to
the first version I stated.
But now, there are
some more quantifiers.
So for every r,
there exists an N.
So why are these two versions
equivalent to each other?
So it's not too hard to
deduce their equivalence.
So let me do that now.
The fact that the
finitary version
implies the infinitary
version claims
should be fairly obvious.
So once you know the
finitary version,
if you give me a coloring of
the positive integers, well
I just have to look
far enough up to this N
and I get the conclusion I want.
But now, in the
other direction--
so in the other direction--
suppose I fix to this r.
So, OK, so I assume
the infinitary version.
I wanted to deduce
the finitary version.
So I start with this r.
And let's suppose the
conclusion were false.
So supposed the
conclusion were false,
namely for every N there
exists some coloring--
so for every N there
exists some coloring--
which we will call
phi sub N, that
avoids monochromatic solutions
to x plus y equals to z.
So I'm going to use this Chi
for shorthand for monochromatic.
So suppose there
exists such a coloring.
And now, I want to take
this collection of colorings
and produce for you a coloring
of the positive integers.
And you can do this basically
by a standard diagonalization
trick.
Namely, we see that by taking
an infinite subsequent,
such that--
so let me call this infinite
sub-sequence phi of--
phi sub-- well, so it's infinite
sub-sequence of this phi sub N,
such that phi sub N
of k stabilizes along
the sub-sequence for every k.
OK, so you can do this simply
by diagonalization trick.
And then, we see that
along the sub-sequence phi
N converges point-wise to some
coloring of the entire set
of positive integers.
And this coloring avoids
monochromatic solutions
to x plus y equals to
z, because if there
were monochromatic
solutions in this coloring
of the entire integers,
then I can look back
to where that came from.
And that would have been the
monochromatic solution in one
of my phi N's.
So this is an
argument that shows
the equivalence between the
finitary form and infinitary
form.
But now, when we look
at the finitary form,
you can ask additional
questions, such as,
how big does this N have
to be as a function of r.
It turns out those kind
of questions in general
are very difficult. And
we know some things.
For this type of questions,
we know some bounds usually.
But the truth is
usually unknown.
And there are
major open problems
in combinatorics of this type.
So there's still a lot
that we do not understand.
OK, so now we have Schur's
Theorem in this form.
Let me show you how to deduce
his conclusion about ruling out
this approach to proving
Fermat's Last theorem.
The claim is the following that
if you have a positive integer
n, then for all
sufficiently large primes p,
there exists x, y,
and z, all belonging
to integers up to p minus 1,
such that their n-th powers
add up like this.
So it's a solution to
Fermat's equation mod p.
All right, so how
can we deduce this
from what we said
about coloring?
So what is the coloring?
OK, so here's what Schur
did, so proof assuming
for now Schur's theorem.
So let's look at the
multiplicative group
of non-zero residues, mod p.
So we know it's a cyclic group
because there's a generator.
So there's a primitive
root generator.
Let H denote the
subgroup of n-th powers.
Well, H is a pretty
big subgroup.
So what's the index of H in
this multiplicative group?
It's at most M.
So think about representing
this as a cyclic group using
a generator.
So H then would be all the
elements whose exponent
is divisible by M. So
this the index is at most
M. It could be smaller.
But it's at most M.
And so in particular,
I can use the H cosets
to partition the multiplicative
group of non-zero residues.
And this is a color.
Virtual partition is the
same thing as a coloring.
There is a bounded
number of colors.
But I let peek at large.
So by Schur's theorem if
p is sufficiently large,
then one of my cosets
should contain a solution
to x plus y equals to z.
What does that look like?
So that one coset, one H coset,
course that contains x, y, z,
such that x plus y
equals to z as integers.
They belong to the same coset.
So x, y, and z belong
to some coset of H,
which means then that x equals
to a times n-th power with a y
equals to a times some
n-th power and little z
equals to a times
some n-th power.
You have this equation.
Put them together.
So that is true.
So now mod p, I
can cancel the a's.
And this produces a
non-trivial solution
to Fermat's equation, mod p.
OK, so this was the
proof of this claim
that this method does
not work for solving
Fermat's Last Theorem.
But, you know, we assumed
this claim of Schur's theorem
that every finite coloring
of the positive integers
contains a monochromatic
solution to x plus y
equals to z.
So we still need to
prove that claim.
So we still need to prove
this combinatorial claim.
And so that's what
we're going to do now.
This is where graph
theory comes in.
So let me state a very
similar-looking theorem
about graphs.
And this is known
as Ramsey's theorem,
although Ramsay's theorem
actually historically came
after Schur's theorem, but
Ramsey's theorem, here, we're
going to use it specifically
in the case for triangles.
So what does it say?
That if you give me an
r, the number of colors,
then there exists
some large N such
that if the edges of the
complete graph, K sub N,
along N vertices are
colored using r colors,
then there exists a
monochromatic triangle
somewhere.
Any questions so far about
any of these statements?
So let's see how Ramsay's
theorem for triangles
is proved.
By the way, I want to give you
a historical note about Frank
Ramsey.
So he's someone who made
significant contributions
to many different areas,
not just in mathematics.
So he contributed to seminal
works in mathematical logic
where this theorem came
from, but also to philosophy
and to economics before his
untimely death at the age of 26
from liver-related problems.
So he's someone whose very short
life contributed tremendously
to academics.
So let's see how Ramsay's
theorem, in this case,
is proved.
We'll do induction on
r, the number of colors.
So for every r, I need
to show you some N,
such that the statement is true.
In the first case, when r equals
to 1, there's not much to do.
Just one color, if I
just have three vertices,
that already is OK.
Three vertices, that's already
a monochromatic triangle.
So from now on, let
r be at least 2.
And suppose the claim
holds for r minus 1 colors,
with N prime being the
corresponding number
of vertices with
r minus 1 colors.
So now let me pick
an arbitrary vertex.
So pick an arbitrary vertex
v and look what happens.
So here's v.
And let me look at
the outgoing edges.
So we'll show that
N being r crimes
N prime minus 1 plus 2 works.
So now, we have a lot
of outgoing edges.
In particular, we
have r times N prime
minus 1 plus 1 outgoing edges.
So by the pigeonhole
principle, some color--
so there exists at least
N prime outgoing edges
with the same color,
let's say, yellow.
So suppose yellow is
the outgoing color.
And let me call
the set of vertices
on the other end
of these edges v0.
So now let's think about
what happens in v0.
So in v0, either v0 contains
a yellow edge, in which case
you get a yellow triangle.
Or we lose the color inside v0.
So the number of
colors goes down.
Else v0 has at most
r minus 1 colors.
And v0 has at least N
prime number of vertices.
So by induction, v0 has
a monochromatic triangle
in the remaining colors.
So that completes the proof of
Ramsay's theorem, in this case,
for triangles.
And if you wish to find out
what is the bound that comes out
of this argument, well,
you can chase to the proof
and get some bound.
The remaining question
now is, what does this all
have to do with Schur's theorem?
So so far, we've talked
about some number theory.
We've talked about
some graph theory
and how to link these
two things together.
And I think this
is a great example.
It's a fairly simple
example, which
I'm about to show you of how to
link these two ideas together.
And this connection,
we'll see many times
in the rest of this course.
I don't want to erase
Schur's theorem.
So let me--
So let's prove Schur's theorem.
So let's start with a coloring.
So let's start with the
coloring of 1 through N.
And I want to form
a graph with colors
on the edges that
are somehow derived
from this coloring
on these integers.
And here's what I'm going to do.
So let's color the
complete graph-- let's
color the edges of the complete
graph on the vertex set
having N plus 1 vertices,
labeled at integers
up to positive integers
up to N plus 1.
But by the Ramsey result we just
proved, if N is large enough,
then there exists a
monochromatic triangle.
So what does it look like?
So let me draw for you a
monochromatic triangle.
Suppose it-- so I haven't told
you what the coloring is yet.
So the coloring
is that I'm going
to color the edge
between i and j,
using the color derived by
applying phi to the number
j minus i, namely the
length of that segment
if I lay out all the
vertices on the number line.
So now have an r coloring
of this complete graph.
So Ramsey tells us
that there exists
a monochromatic triangle.
The triangle sits on
vertices i, j, and k.
And the rule tells us that the
colors are phi of k minus i,
phi of j minus i,
and phi of k minus j.
So these three numbers,
they have the same coloring.
But, look, if I set these
numbers to be x, y, and z--
so x being j minus
i, for instance--
then x plus y equals to z.
And they all have
the same color.
So this monochromatic
triangle gives us
a monochromatic
equation, 2x plus y
equals to z, thereby concluding
the proof of Schur's theorem.
OK, so this rounds out the
discussion for now of--
well, we started with some
statement about number theory.
And then we took this
detour to graph theory,
looking at Ramsay's theorems
of monochromatic triangles,
and then go back
to number theory
and proved the result
that Schur did.
So how does go to graphs help?
So why was this advantageous?
What do you guys think?
So I claim that by
going to graphs,
we added some extra flexibility
to what we can play with.
For example, we started
out with a problem
where there were only
N things being colored.
And then we moved to
graphs where about--
well, N choose 2 or N squared
objects are being colored.
And then we did an
induction argument.
So remember in the proof of
Ramsey's theorem up there,
there was an induction
argument taking all vertices.
And that argument doesn't
make that much sense
if you stayed
within the numbers.
Somehow moving to
graphs gave you
that extra flexibility
allow you to do more things.
And this is one
of the advantages
of moving from
problem about numbers
to a problem about graphs.
And we'll see this
connection later on as well.
Yeah?
AUDIENCE: Sort of
related to that.
Are there better bounds
known for this specific,
like Schur's result
of that power on e,
because the N's here
would be pretty bad.
YUFEI ZHAO: Right, so Ashwan
asked, so what about bounds?
So what do we know about bounds?
So I don't know off the
top of my head the answers
to those questions.
But in general,
they're quite open.
So there are exponential gaps
between lower and upper bounds
on our knowledge of what
is the optimal N you
can put in the theorem.
Any more questions?
All right so, I think this is
a good point for us to-- so
usually when I give
90-minute lectures,
I like to take a short
2-minute break in between.
So I want to do that.
And then in the
second half, I want
to take you through a tour
of additive combinatorics.
So tell you about some of
the modern developments.
Now, this is an exciting
field where it started out,
I think, roughly with Schur's
theorem that we just discussed.
That started about
100 years ago.
But a lot has taken place
in the past century.
And there's still a lot of
ongoing exciting research
developments.
So in the second
half of this lecture,
I want to give you a tour
through those developments
and show you some
of the highlights
from additive combinatorics.
So let's take a
quick 2-minute break.
And feel free to ask
questions in the meantime.
So another part of
the writing assignment
in addition to course
notes is a contribution
to Wikipedia, which is, you
know, nowadays, of course,
you know, if you hear
some word like Szemeredi
's regularity lemma
the first thing you do
is type into Google.
And more often than not the
first link that comes up
is Wikipedia.
And, you know, some
of the articles,
they are all right, and some of
them are really not all right.
And it would be fantastic
for future students
and also for yourselves
if there were better entry
points to this area by having
higher quality Wikipedia
articles or articles
that are simply
missing about specific topics.
So one of the assignments--
again, this can
be collaborative.
So I'll give you
more information
how to do that later--
is to contribute to
Wikipedia and roughly
contribute one high
quality article
or edit some
existing articles so
that they become high quality.
Yep.
AUDIENCE: Can we
something similar to LMDB
with creating a website
that has all the information
needed in combinatorics?
YUFEI ZHAO: So we
can talk about that.
So if there are other
ideas about how to do this,
we can definitely open
the chatting about that.
So the other thing
is that instead
of holding the usual office
hours, what I like to do is--
so this class ends at 4:00 PM.
So after 4:00, I'll go
up to the Math Common
Room, which is
just right upstairs
and hang out there for a bit.
If you have questions, you
want to chat, come talk to me.
I'd be happy to chat about
anything related or not related
to the course.
And before homeworks
are due, I will
try to set up some special
office hours for you
in case you want to ask
about homework problems.
And if you want to meet
with me individually,
please just send me an email.
Oh, one more thing
about the course notes.
So because I want to
do quality control,
so here is the process that will
happen with the course notes.
So the first lecture
is already online.
So you can already see.
So I've written up the lecture
notes for the first lecture.
And you can use
that as an example
of what I'm looking for.
So I'm looking for
people to sign up
starting from the
next lecture, and I
will send out a link tonight.
For future lectures, so
whoever writes the lecture,
I'll the lecture, and
then within one day, so
by the end of the day
after the lecture,
it will be good if
they were already
at least some sketch,
some rough draft at least
containing the
theorem statements
and whatnot from
the day's lecture.
So that the next person can
start writing afterwards.
But once you are
done, once you feel
that you have a polished version
of the lecture, write up,
ideally within four
days of the lecture--
so that in terms of
expectations and timelines,
again all of this
information is online--
so you're finished with
polishing your lecture notes,
within four days send me an
email, so both co-authors
if there are two of
you, and I will schedule
an appointment,
about half an hour,
where I will sit
down with you to go
through what you've written
and tell you some comments.
So you can go back
and polish it further.
And hopefully, that will
just be a one round thing.
If more rounds are needed,
well, it's not ideal,
but we'll make it happen until
the notes are ready to use
for future generations.
OK, any questions about any
of the course logistics?
All right, so in the second
half of today's lecture,
I want to take you through
a tour of modern additive
combinatorics.
And this is an area
of research which
I am actively involved in.
And it's something that
I am quite excited about.
And part of the reason
why I teach this course--
this course is something that
I developed a couple years ago
when I taught for
the first time then--
because I want to
introduce you guys
to this very active and
exciting area of research.
Now, what is added
combinatorics?
The term itself is
actually fairly new.
So the term, additive
combinatorics, I believe
was coined by Terry Tao back
in the early 2000s as somewhat
of a rebranding of an area that
already existed, but then got
a lot of exciting developments
in the early 2000s.
It's a deep and far reaching
subject with many connections
to areas like graph theory,
harmonic analysis, or Fourier
analysis, ergodic theory,
discrete geometry, logic
and model theory, and has many
connections all over the place,
and also has many deep theorems.
So let me take you through a
tour historically of, I think,
some of the major
milestones and landmarks
in additive combinatorics.
So after Schur's theorem,
which we discussed
in the first half
of today's lecture,
the next big result I would say
is Van der Waerden's theorem,
which was 1927.
Van der Waerden's theorem
says that every coloring
of the positive integers
using finite many colors
contains arbitrarily long
arithmetic progressions.
So we'll see arithmetic
progressions come up a lot.
So from now on we'll
abbreviate this word by AP.
So AP stands for
Arithmetic Progressions.
So instead of Schur's
theorem where you just
find a single solution
to x plus y equals to z,
so now, we're finding a
much bigger structure.
Keep in mind, so a
novice mistake people
make is to confuse arbitrarily
long arithmetic progressions
with infinitely long.
So these are definitely
not the same.
So you can think about.
I'll leave it to you as
an exercise, well, also
homework exercise, that you can
color the integers with just
two colors in a
way that destroys
all possible infinitely long
monochromatic arithmetic
progressions.
So arbitrarily long is very
different from infinitely long.
Now, so this was a great result,
but it provokes more questions.
So Erdos-Turan in the
'30s, they asked--
well, they conjectured
that the true reason in Van
der Waerden's theorem of having
long arithmetic progressions,
it's not so much
that you're coloring.
It's just because if you
use finitely many colors,
then one of the color classes
must have fairly high density.
So one of the classes if you use
r colors has density at least 1
over r.
And they conjectured
that every subset
of the positive integers, or the
integers with positive density,
contains long--
so arbitrarily long
arithmetic progressions.
You may ask, what
does it mean, density?
So you can define density
in many different ways.
And it doesn't
actually really matter
that much which
definition you use.
But let me write
down one definition.
So you can define given
a subset of integers
the upper density, or
rather, let me just
say that it has positive
upper density, if when we take
the lim sup as n goes to
infinity and look at we'll
take a scaling window and look
at what fraction of that window
is a, then this number,
this limit sup is positive.
So that's one definition
of positive density.
There are many other
definitions, sometimes known
as the Banach density.
And you can take variations.
I mean, for the purpose
of this discussion,
they're all roughly equivalent.
So let's not worry too
much about which definition
of density we use here.
All right, so Erdos
and Turan conjectured
that the true reason for
Van der Waerden's theorem
is that one of the color
classes has positive density.
And this turned out to be an
amazingly prescient question
and that one had to
wait several decades.
So this conjecture was
made in the '30s, in 1936.
So you had to wait several
decades before finding out
what the answer is.
So in a foundational
theorem, in the subject
known as Roth's theorem--
so Roth proved it in the '50s.
I think '53--
Roth proved that, I
think, '53, in the '50s--
that k equals to 3 is true.
So if I say that it
contains k term, arithmetic
progressions for every k.
And Roth proved that
every positive density
subset contains a 3-term
arithmetic progression.
And already, Roth introduced
very important ideas
that we will see in this
course in two different forms.
So in the first half
the course, we'll
see a graph theoretic
proof that was found later
in the '70s of Roth's theorem.
And then in the
second half, we'll
see Roth's original proof
that used Fourier analysis.
So Fourier analysis
in number theory
is also known as the
Hardy-Littlewood circle method.
It's a powerful method in
analytic number theory.
But there are very interesting
new ideas introduced by Roth as
well in developing this result.
The full conjecture was
settled by Szemeredi.
It took another
couple of decades.
So in the late '70s, Szemeredi
proved his landmark theorem
that confirmed the
Erdos-Turan conjecture.
Szemeredi's theorem
is a deep theorem.
So this theorem
is the proof, what
the original combinatorial
proof is a tour de force.
And you can look
at the introduction
of his paper, where there is
an enormously complex diagram--
so you can see this
in the course notes--
that lays out the logical
dependencies of all the lemmas
and propositions in his paper.
And even if you assume every
single statement is true,
looking at that diagram,
it's not immediately clear
what is going on because
the logical dependencies are
so involved.
So this was a really
complex proof.
But not only that, Szemeredi's
theorem actually motivated
a lot of subsequent research.
So later on, researchers
from other areas
came in and found also
sophisticated proofs
of Szemeredi's theorem
from other areas
and using other
tools, including--
and here are some of the
most important perspectives,
later perspectives, of
Szemeredi's theorem.
So there was a proof
using ergodic theory
that followed fairly
shortly after Szemeredi's
original proof.
This is due to Furstenberg.
And initially, it wasn't clear,
because all of these proofs
were so involved.
It wasn't clear if the ergodic
theoretic proof was genuinely
something new, or it was a
rephrasing of Szemeredi's
combinatorial proof.
But then very quickly
it was realized
that there were extensions
of Szemeredi's theorem,
other combinatorial results
that the ergodic theorists could
establish using their methods,
so using the same methods
or extensions of
the same methods
that combinatorialists
did not know how to do.
And to this date,
there are still
theorems for which
the only known proofs
use ergodic theory,
so extensions
of Szemeredi's theorem.
And I will mention
one later on today.
So that's one of
the perspectives.
The other perspective that
was also quite influential
there is something known
as higher order Fourier
analysis, which was pioneered
by Tim Gowers' in around 2000.
So Gowers won the
Fields Medal, party
for his work on
Banach spaces but also
party for this development.
So higher order Fourier
analysis is in some sense
an extension of Roth's theorem.
So anyway, Roth also
won a Fields Medal,
although this is not
his most famous term.
I'll say his second
most famous theorem.
So Roth used this
Fourier analysis
in the sense of Hardy-Littlewood
to control 3-term arithmetic
progressions.
But it turns out
that that method
for very good
fundamental reasons
completely fails for 4-term
arithmetic progressions.
So we'll see later in the
course why that's the case,
why is it that you cannot do
Fourier analysis to control
4-term APs.
But Gowers managed to find a
way to overcome that difficulty.
And he came up
with an extension,
with a generalization
of Fourier analysis,
very powerful, very
difficult to use, actually.
But that allows you to
understand longer arithmetic
progressions.
Another very
influential approach
is called hypergraph regularity.
So the hypergraph
regularity method
was also discovered in the early
2000s independently by a team
led by Rodl and also by Gowers.
So the hypergraph
regularity method
is an extension of what's known
as Szemeredi's regularity,
Szemeredi's graph
regularity method.
And this is the
method that will be
a central topic in the
first half of this course.
And it's a method that is
quite central, or at least some
of the ideas quite central,
to Szemeredi's method.
And he gave an
alternative proof.
He and Ruzsa gave an alternative
proof of Roth's theorem
using graph theory.
And for a long time,
people realized
that one could extend some of
those ideas to hypergraphs.
But working out how
that proof goes actually
took an enormous amount
of time and effort
and resulted in this amazing
theorem on hypergraph.
Let me mention these are
not the only methods that
were used to extend
Szemeredi's theorem
or give alternate proofs.
There are many others.
For example, you may
have heard of something
called the polymath project.
Raise your hand if you heard
of the polymath project.
OK, great.
So maybe about half of you.
So this is an online
collaborative project
started by Tim Gowers and also
famous people like Terry Tao.
And they were all quite involved
in various polymath projects.
And the first successful
polymath project
produced a combinatorial
proof of something
known as the density
Hales-Jewett theorem.
So I won't explain
what it this here.
So it's something which
is related to tic tac toe.
But let me not go into that.
So it's a deep combinatorial
theorem that had they
known earlier using
ergodic theoretic methods,
but they gave a new
combinatorial proof,
in particular gave some
concrete bounds on this theorem
and that in particular also
implies Szemeredi's theorem.
So this gave a new proof.
And as a result, they--
it's an online
collaborative project--
so they published this paper
under the pseudonym DHJ
Polymath, where DHJ stands
for Density Hales Jewett.
And they kept the
same name for all
of the subsequent
papers published
by the polymath project.
So as you see through
all of these examples
that there a lot of
work that were motivated
by Szemeredi's theorem.
This is truly a
foundational result,
a foundational
theorem that gave way
to a lot of important research.
And Szemeredi himself
received an Apple Prize
for his seminal contributions
to combinatorics and also
theoretical computer science.
We still don't
understand in some sense
completely what
Szemeredi's theorem--
you know, for example, we do
understand the optimal bounds.
And also more importantly,
conceptually, we
don't really understand
how these methods are
related to each other.
So there's some vague
sense that they all
have some common things.
But there is a lot of mystery as
to what do these methods coming
from very different areas--
ergodic theory,
harmonic analysis, you
what do they all have
to do with each other
but there is central theme.
And this is also going to be
a theme in this course, which
goes under the name--
and I believe Terry Tao is the
one who popularized this name--
the dichotomy between structure
and randomness, structure
and pseudo randomness.
Somehow it's a really fancy way
of saying signal versus noise.
So I give you some object, I
give you some complex object,
and there is some
mathematical way
to separate the structure
from some noisy aspects, which
behave random-like.
So there will be many
places in this course
where this dichotomy will
play an important role.
Any questions at this point?
I want to take you through some
generalizations and extensions
of Szemeredi's theorem.
So first, let's
look at what happens
if we go to higher dimensions.
Suppose we have a
subset in D dimensions,
d-dimensional lattice.
So we can also define
some notion of density.
Again, it doesn't
matter precisely what
is the notion you use.
For example, we can say that
it has a positive upper density
if this lim sup is positive.
So Szemeredi's theorem
in one dimension
tells us that if you have some
sort of positive density, then
I can find arbitrarily long
arithmetic progressions.
So what should the
corresponding generalization
in higher dimensions?
Well, here's a notion
that I can define, namely
that we say that a contains
arbitrary constellations
to mean that--
so what does that mean?
So a constellation,
you can think of it
as some finite pattern, so a
set of stars in the sky, so
some pattern.
And I want to find that
pattern somewhere in a, where
I'm allowed to dilate.
So I'm allowed to do to
multiply pattern by some number
and also translate.
So on the finite pattern--
so what I mean precisely is
that for every finite subset
of the grid, there exists some
translation and some dilation,
such that once I apply this
dilation and translation
to my pattern F, meaning I'm
looking at the image of this F
under this transformation,
then this set lies inside a.
So you see that
arithmetic progressions
is the constellation,
just numbers 1 through k.
So that's a definition.
And the multi-dimensional
Szemeredi's theorem--
so the multi-dimensional
generalization
of Szemeredi's theorem says
that for every subset--
so every subset of the
d-dimensional lattice
of possible density contains
arbitrary constellations.
You give me a pattern,
and I can find
this pattern inside a, provided
that a has positive density.
So in particular, if I want to
find a 10 by 10 square grid,
so meaning suppose I want to
find a pattern which consists
of something like that, a
10 by 10 square grid, where
all of these lengths
are equal, but I
don't specify what they are.
But as long as they are equal,
then the theorem tells me
that as long as a
has positive density,
then I can find such
a pattern inside a.
So this theorem was proved
by Furstenberg and Ketsen.
So you see that it is a
generalization of Szemeredi's.
So the one-dimensional case is
precisely Szemeredi's theorem.
So Furstenberg and Ketsen,
using ergodic theory
showed that one can
generalize Szemeredi's theorem
to the multi-dimensional
setting.
However, the
combinatorial approaches
employed by Szemeredi did
not easily generalize.
So it took another couple
of decades at least
for people to find a
combinatorial proof
of this result. And
namely that happened
with the hypergraph
regularity method.
So this was one of the
motivations of this project.
And you say, OK,
what's the point
of having different proofs?
Well, for one thing it's nice
to know different perspectives
to important theorem.
But there's also
concrete objective.
In particular, it turns out
that if you prove something
using ergodic theory, because--
we will not discuss ergodic
theory in this course.
But roughly, one of the
early steps in such a proof
applies compactness.
And that already
destroys any chance
of getting concrete
quantitative bounds.
So you can ask if I want
to find a 10 by 10 pattern
and I have density 1%, how
large do I need to look?
How far do I have to look in
order to find that pattern?
So that's a
quantitative question
that is actually not at all
addressed by ergodic theory.
So the later methods using
combinatorial methods
gave you concrete bounds.
And so there are some
concrete differences
between these methods.
So this theorem
reminds me of the scene
from the movie a
Beautiful Mind, which
is one of the greatest
mathematical movies
in some sense.
And so there's a scene
there where Russell Crowe
playing John Nash--
so there were at
this fancy party.
And Nash was with his
soon to be wife, Alicia.
And he points to the sky
and tells her, pick a shape.
Pick a shape and I can find
for you among the stars.
And so this is what the
theorem allows you to do it.
So give me a shape
and I can find
that constellation inside a.
Let's look at other
generalizations.
So far, we are looking
at linear patterns.
So we're looking at linear
dilations and translations.
But what about
polynomial patterns?
So here's a question.
Suppose I give you a dense
subset, a positive density
subset of integers.
Can you find two numbers whose
difference is a perfect square?
So this question
was asked by Lovasz.
And a positive answer was given
in the late '70s by Furstenberg
and Sarkozy independently.
So Furstenberg and Sarkozy,
they showed using different
methods-- so one ergodic
theoretic and the other is more
harmonic analytic--
that every subset of the
integers, so every subset
of positive integers,
with positive density
contains two numbers
differing by a perfect square.
So in other words,
we can always find
the pattern x plus y squared.
So what about other
polynomial patterns?
Instead of this y
squared, suppose you just
give me some other
polynomial or maybe
a collection of polynomials.
So what can I say?
Well, there are some things
for which this is not true.
Can you give me an
example where if I
putting the wrong
polynomial it's not true?
What if the polynomial
is the constant 1?
If you take the even
numbers, has density 1/2,
but it doesn't contain any
patterns of x and x plus 1.
So I need to say some hypotheses
about these polynomials.
So a vast generalization
of this result,
so known as polynomial
Szemeredi theorem,
says that if A is a subset of
integers with positive density,
and if we have these
polynomials, P1 through Pk
with integer coefficients
and zero constant terms,
then I can always
find a pattern.
So there exists some x
and positive integer y
such that this pattern, x
plus P1 of y, x plus P2 of y,
and so on x, plus Pk of
y, they all lie in A.
So in other words, succinctly,
every subset of integers
with positive density contains
arbitrary polynomial patterns.
So this was proved--
so this was an
important result proof
by Bergelson and Liebman
using ergodic theory.
And so far for this
general statement,
the only known proof
uses ergodic theory.
So there was some
recent developments,
recent pretty
exciting developments
that for some
specific cases where
if you have some additional
restrictions on the P's, then
there are other methods
coming from Fourier analytic,
harmonic analytic methods
that could give you
a different proof that allows
you to get some bounds.
Remember, the ergodic
proof gives you no bounds.
But so far, in general,
the only method known
is ergodic theoretic.
And actually,
Bergelson and Liebman
proved something which is more
general than what I've stated.
So this is also true in a
multidimensional setting.
I won't state that precisely,
but you can imagine what it is.
Let me mention one more theorem
that many of you I imagine
have heard of.
And this is the
Green Tao theorem.
So the Green Tao theorem
says that the primes
contain arbitrarily long
arithmetic progressions.
So this is a famous theorem.
And it's one of the
most celebrated results
of the past couple of decades.
And it resolved some
longstanding folklore
conjectures in number theory.
The Green Tao
theorem, well, you see
that in form it looks somewhat
like Szemeredi 's theorem.
But it doesn't follow
from Szemeredi 's theorem.
Well, the primes, they
don't have positive density.
The prime number
theorem tells us
that density decays
like 1 over log n.
So what about quantity versions
of Szemeredi 's theorem?
It is possible.
Although we do not know how
to prove such statement,
it is possible that
a density of primes
alone might guarantee the
Green Tao theorem in that it
is possible that
Szemeredi 's theorem
is true for any set
whose density decays
like the prime numbers,
like 1 over log n.
But no we're quite far from
proving such a statement.
And that's not what
Green and Tao did.
Instead, they took Szemeredi
's theorem as a black box
and applied it to some
variant of the primes
and showed that
inside this variant,
Szemeredi 's theorem
is also true,
and that the primes sit inside
this variant of the primes,
known as pseudo primes, as a set
of relatively positive density,
but somehow
transferring Szemeredi
's theorem from the dense
setting to a sparser setting.
So this is a very
exciting technique.
And as a result, Green-Tao
proved not just that the primes
contain arbitrarily long
arithmetic progressions,
but every relatively dense,
so relatively positive
density subset, of the
primes contains arbitrarily
long arithmetic progressions.
To prove this theorem
they incorporated
many different ideas coming
from many different areas
of mathematics, including
harmonic analysis,
some ideas coming from
combinatorics, and number
theory as well.
So there were some innovations
at the time in number theory
that were employed
in this result.
So this is certainly
a landmark theorem.
And although we will
not discuss a full proof
of the Green-Tao theorem, we
will go into some of the ideas
through this course.
And I will show
you bits and pieces
that we will see
throughout the course.
So this is meant to be
a very fast tour of what
happened in the last 100 years
in additive combinatorics,
taking you from Schur's
theorem, which was really
about 100 years
ago, to something
that is much more modern.
But now, instead of
being up in the stars,
let's come back down to Earth.
And I want to talk about
what we'll do next.
So what are some of the
things that we can actually
prove that doesn't
involve taking up 50 pages
using a complex logical diagram,
as Szemeredi did in his paper.
So what are some of the simple
things that we can start with?
Well, so first, let's go
back to Roth's theorem.
So Roth's theorem, we
stated it up there.
But let me restate it
in a finitary form.
So Roth's theorem
is the statement
that every subset of
integers 1 through n that
avoids 3-term
arithmetic progressions
must have size little
o of N. So earlier we
gave an infinitary
statement that if you
have a positive density
subset of the integers
that it contains
a three AP, this
is an equivalent
finitary statement.
Roth's original proof
used Fourier analysis.
And a different proof was
given in the '70s by Rusza
and Szemeredi using
graph theoretic methods.
So how does graph theory
have to do with this result?
And this shouldn't be
surprising to this point, given
that we already saw how we
used Ramsay's theorem, graph
theoretic result, to
prove Schur's theorem,
which is something that
is number theoretic.
So something similar happens.
But now, the question is what
is the graph theoretic problem
that we need to look at?
So for Schur's theorem it was
Ramsey's theorem for triangles.
But what about for
Roth's theorem?
A naive guess is the following.
So what's the question
that we should ask?
Here's a somewhat naive
guess, which turns out
not to be the right
question, but still
an interesting
question, which is
what is the maximum number of
edges in a triangle-free graph
on n vertices?
Now, this is not
totally a stupid guess,
because as you imagine from what
we said with Schur's theorem,
somehow you want to
set up a graph so
that the triangles correspond
to the 3-term arithmetic
progressions.
And you want to set
it up in such a way
that this question about what's
the maximum size subset of 1
through n without
3 APs translates
into some question about what's
the maximum number of edges
in a graph that
has some property?
So what is that property?
So this is not a
totally stupid guess.
But it turns out this
question is relatively easy.
Still it has a name.
So this was found by
Mantel about 100 years ago,
so known as Mantel's theorem.
And the answer, well,
we'll see a proof.
So the first thing we'll
do in the next lecture
is prove Mantel's theorem, but
I do want to hold suspense.
I mean the answer,
it turns out to be
fairly simple to describe.
Namely that you split
the vertices into two
basically equal halves.
And you join all the possible
edges between the two halves.
So this complete bipartide graph
with two equal-sized parts.
And it turns out this graph,
you see this triangle-free
and also turns out to have
the maximum number of edges.
Yeah, question.
AUDIENCE: What are asymptotics
for three arithmetic
progression of--
YUFEI ZHAO: Let me get
to that in a second.
OK, so I'll talk about
asymptotics in a second.
So it turns that this is not the
right graph theoretic question
to ask.
So what is the right graph
theoretic question to ask?
I'll tell you what it is.
I mean it shouldn't be
clear to you at this point.
It still seems like an
interesting question,
but it's also somewhat bizarre
to think about if you've never
seen this before.
So what is the maximum
number of edges
in an n vertex graph, where
every edge lies in exactly one
triangle?
So I want a graph with
lots and lots of edges
where every edge sits
in exactly one triangle.
Now, you might have
some difficulty even
coming up with good graphs
that have this property.
And that's OK.
These are very strange
things to think about.
But we'll see many
examples of it later on.
We'll also see
how Roth's theorem
is connected to this
graph theoretic question.
Just to give you
a hint, you know,
where does exactly one
triangle come from,
it's because even if you avoid
3-term arithmetic progressions,
there are still these
trivial 3-term arithmetic
progressions, where you keep
the same number three times.
And in graph
theoretic world, that
comes to the unique triangle
that every edge sits on.
So to address the question
about quantitative bounds,
for Roth's theorem, it
turns out that we have
upper bounds and lower bounds.
And it is still a
wide open question
as to what these
things should be.
And roughly speaking,
the best lower bound
comes from a construction,
which we'll see later
in this course, the higher size
around n divided by e to the c
root log n.
And the best upper bound is of
the form roughly n over log n.
That's maybe a little
bit hard to think
about how these numbers behave.
So if you raise both
sides to-- the denominator
to e to the something, then
it's maybe easier to compare.
But it's still a pretty far gap.
So still a pretty big gap.
There's a famous conjecture
of Erdos some of you
might have heard
of, that if you have
a subset of the
positive integers
with divergent
harmonic series, then
it contains arbitrarily
long automatic progressions.
That's a very
attractive statement.
But somehow I don't like
the statement so much,
because it seems to
make a too pretty.
And the statement
really is about what
is the bounds on Roth's theorem
and on Szemeredi 's theorem.
And having divergent
harmonic series
is roughly the same as trying
to prove Roth's theorem slightly
better than the bound
that we currently have,
somehow breaking this
logarithmic barrier.
So that conjecture, that having
divergent harmonic series,
implies 3-term
APs is still open.
That is still open.
So where the bound's very
close to what we can prove,
but it is still open.
For this question, we will
see later in this course,
once we've developed Szemeredi
's regularity lemma that we
can prove an upper bound of o
to the n squared, so little n.
And that will suffice for
proving Roth's theorem.
It turns out that
we don't know what
the right answer should be.
We don't know what is
the best such graph.
And it turns out the
best construction
for this graph there comes from
over here, the best lower bound
construction of a
set, of a large set
without 3-term
arithmetic progressions.
So I'm giving you
a preview of more
of these connections between
additive combinatorics on one
hand and graph theory
on the other hand
that we'll see
throughout this course.
Any questions?
OK.
So just to tell you what's
going to happen next,
so the next thing that
we're going to discuss
is basically extremal
graph theory.
And in particular, if you
forbid some structure,
such as a triangle, maybe a four
cycle, maybe some other graph,
what can you say about the
maximum number of edges?
And there are still a lot of
interesting open problems,
even for that.
I forbid some H. What's the
maximum number of edges?
So the next few lectures
will be on that topic.
