# Cup Sets, Sunflowers, and Matrix Multiplication

This post follows a recent paper On sunflowers  and matrix multiplication by Noga Alon, Amir Spilka, and Christopher Umens (ASU11) which rely on an earlier paper Group-theoretic algorithms for matrix multiplication, by Henry Cohn, Robert Kleinberg, Balasz Szegedy, and Christopher Umans (CKSU05), and refers also to a paper by Don Coppersmith and Shmuel Winograd (CW90).

## Three famous problems

The Erdos-Rado sunflower (Delta system) theorem and conjecture was already menioned in this post on extremal set theory.

A sunflower (a.k.a. Delta-system) of size $r$ is a family of sets $A_1, A_2, \dots, A_r$ such that every element that belongs to more than oneofthe sets belongs to all of them. A basic and simple result of Erdos and Rado asserts that

Erdos-Rado sunflower theorem: There is a function $f(k,r)$ so that every family $\cal F$ of $k$-sets with more than $f(k,r)$ members contains a sunflower of size $r$.

One of the most famous open problems in extremal combinatorics is:

The Erdos-Rado conjecture: Prove that $f(k,r) \le c_r^k$.

Here, $c_r$ is a constant depending on $r$. This is most interesting already for $r=3$.

### Three term arithmetic progressions

The cup set problem (three terms arithmetic progressions in $(Z/3Z)^n$):

The cup set problem was also discussed here quite extensively. (See, e.g. this post.)

Let $\Gamma=$$\{0,1,2\}^n$. The cap set problem  asks for the maximum number of elements in a subset of $\Gamma$ which contains no arithmetic progression of size three or, alternatively, no three vectors that sum up to 0(modulo 3). (Such a set is called a cup set.) If $A$ is a cap set of maximum size we can ask how the function $h(n)=3^n/|A|$ behaves. Roy Meshulam proved, using Roth’s argument, that $h(n) \ge n$. Edell found an example of a cap set of size $2.2^n$. So $h(n) \le (3/2.2)^n$.  The gap is exponential.

The strong cap set conjecture: $h(n) \ge (1+\epsilon)^n$ for some $\epsilon >0$.

Of course, the cap set problem is closely related to

Erdos-Turan problem (for $r=3$): What is the larget size $r_3(n)$ of a subest of {1,2,…,n} without 3-term arithmetic progression?

### Matrix multiplications

Let ω be the smallest real number so that there is an algorithm for multiplying  two $n \times n$ matrices which requires $O(n^\omega )$ arithmetic operations.

The ω=2 conjecture: ω=2.

A very recent breakthrough by Virginia Vassilevska Williams (independently) following an earlier breakthrough by Andrew Stothers improved the Coppersmith-Winograd algorithm which gave ω =2.376, to 2.374 and 2.373 respectively. (See the discussions over Lipton’s blog (1,2), Shtetl optimized, and Computational Complexity.)

It turns out that these three conjectures are related. (The connection of matrix multiplication and the Erdos-Turan problem is fairly old, but I am not sure what an even drastic improvment of Behrends’s lower bound would say about $\omega$.)

## Three combinatorial conjectures that imply ω=2.

Remarkably, an affarmative answer for the ω=2 conjecture would folow from each one of three combinatorial conjectures. One conjecture goes back to CW90 and two were described in CKSU05. I will not present the precise formulations in order to encourage the readers to look at the original papers. (Maybe I will add the formulations later.)

The no disjoint equivoluminous subsets conjecture (CW90).

The Strong UPS conjecture (CKSU05).

Theorem: Conjecture CW90 implies the strong UPS conjecture.

CKSU’s two-family conjecture (CKSU05).

## Relations between these problems

Here are some results taken from ASU11 about the relations between these combinatorial questions. The first result goes back to Erdos and Szemeredi.

The weak sunflower conjecture: A family $\cal F$ of subsets of {1,2,…,n}  with no sunflower of size 3 can have at most $(2-\epsilon)^n$ sets.

The following results are not difficult.

Theorem: The strong sunflower conjecture implies the weak sunflower conjecture.

Theorem: The strong cup set conjecture also implies the weak sunflower conjecture.

Theorem: The weak sunflower conjecture implies that the CW90 conjecture is false.

It follows that CW90 conjecture is in conflict both with the Erdos Rado sunflower conjecture and with the strong cup set conjecture.

Theorem: The strong cup set conjecture implies that the strong UPS conjecture is false.

While two family theorems are quite popular in extremal combinatorics (see this post and this one), CKSU’s two family conjecture is still rather isolated from other combinatorics.

## What to believe?

This is a nice topic for discussion.

# The Cap-Set Problem and Frankl-Rodl Theorem (C)

Update: This is a third of three posts (part I, part II) proposing some extensions of the cap set problem and some connections with the Frankl Rodl theorem. Here is a post presenting the problem on Terry Tao’s blog (March 2007). Here is an open discussion post around Roth theorem (March 2009). Here are two “open discussion” posts on the cap set problem (first, and second) from Gowers’s blog (January 2011). These posts include several interesting Fourier theoretic approaches towards improvement of the Roth-Meshulam bound. The second post describes a startling example by Seva Lev.

Suppose that the maximum size of a cap set in $\Gamma(n)=\{1,2,3\}^n$ is $n/g(n)$.  Here is an obvious fact:

The maximum size of a set $F$ in $G$ with the property that every $x,y\in G$ ($x \ne y$) satisfy $x+y \notin G$ is at most the maximum size of a cap set in $G$

Proof: Indeed the condition for $F$ is stronger than being a cap set. We require for every $x,y \in F$ $x \ne y$ not only that $x+y \notin F$ but even $x+y \notin G$.

## Part C.  A more direct relation between Frankl-Rodl’s theorem and the cap set problem and all sorts of fine gradings on subsets of {1,2,…,n}.

In part A we moved from sets avoiding affine lines (cap sets) to sets avoiding “modular lines” and used the Frankl-Rodl theorem to deal with the latter. This may look artificial. Here is a simpler connection between the cap set problem and the Frankl-Rodl theorem.

# Around the Cap-Set problem (B)

## Part B: Finding special cap sets

This is a second part in a 3-part series about variations on the cap set problem that I studied with Roy Meshulam. (The first post is here.)  I will use here a different notation than in part A to make it compatible with various posts of polymath1 which consider similar objects. We write $\Gamma(n) = \{0,1,2\}^n$ and $\Gamma_{a,b,c}$ denotes all vectors of length $n$ with $a$ 0’s, $b$ 1’s and $c$ 2’s.

An affine line are three vectors $x,y,z$ so that $x_i + y_i + z_i = 0 (mod~3)$ for every $i$. A cap set is a subset of $\Gamma(n)$ without an affine line. The cap set problem asks for the maximum size of a cap set. For information on the cap set problem itself look here.  For questions regarding density of sets avoiding other types of lines look here.

In the first part we considered the case that $n=3m$ and considered “modular lines” , i.e., three vectors  $x,y,z$ so that the number of solutions to $x_i+y_i+z_i=a(mod~3)$  is divisible by $m$ for every $a$. When $n$ is divisible by 9 the Frankl-Rodl theorem implies that a subset of $\Gamma$ with more than $(3-\epsilon)^n$ elements must contain a modular line, and even a modular line with two equal elements. (Question: does the same statement follow from Frankl-Rodl theorem when $n$ is divisible by 3 and not by 9?)

### 10. Some propositions in the desired direction.

Proposition 6:  Suppose that $n=3m$, and that $m=1 (mod~3)$. If every set of size > $(3-\epsilon)^n$ has a modular line, then every set of size > $(3-\epsilon)^n$ has a modular line which is either an affine  line, or  the number of $i$s such that $x_i + y_i + z_i =a$ is precisely $m$ for $a=0,1,2$.

(So we can get rid of a few “types” of modular lines which are not really affine  lines, but not all of them.)

Proposition 6 follows from the much more transparent  proposition:

Proposition 7:  There are sets of constant density which do not contain a modular line $\{x,y,z\}$ for which the numbers of solutions of $x_i + y_i + z_i =a(mod~3)$ for $a=0,1,2$ is never a permutation of $(0,m,2m)$.

Proof: Consider all vectors where the sum of the coordinates is zero modulo 3. This property is preserved under sums but it does not hold for the sums we are looking at. (All the quantities: 0+m+4m, 0+0 +4m, 0+2m+2m, 0+2m+0, m, 2m are not divisible by 3.)

### 11. A strong variant form of the problem

Here are innocent-looking variants of Problem 1: (the existence of modular lines)

Problem 6:  How large can a set $F$ in $\Gamma(7m)$  be without three vectors x, y and z such that $|\{i:x_i+y_i+z_i=a\}|=o(mod~m)$  for $a=0,1,2,3,4,5,6$?

Problem 7:  How large can a set F in $\Gamma (7m)$  be without three vectors x, y and z such that $|\{i: x_1+y_i+z_i=a\}| = m$ for every a=0,1,2,3,4,5,6? Continue reading

# An Open Discussion and Polls: Around Roth’s Theorem

Suppose that $R_n$ is a subset of $\{1,2,\dots, n \}$ of maximum cardinality not containing an arithmetic progression of length 3. Let $g(n)=n/|R_n|$.

How does $g(n)$ behave? We do not really know. Will it help talking about it? Can we somehow look beyond the horizon and try to guess what the truth is?

Update 1: for the discussion: Here are a few more specific questions that we can wonder about and discuss.

1) Is the robustness of Behrend’s bound an indication that the truth is in this neighborhood.

2) Why arn’t there analogs for Salem-Spencer and Behrend’s constructions for the cup set problem?

3) What type of growth functions can we expect at all as the answer to such problems?

4) Where is the deadlock in improving the upper bounds for AP-free sets?

5) What new kind of examples one should try in order to improve the lower bounds? Are there some directions that were already extensively explored?

6) Can you offer some analogies? Other problems that might be related?

# Frankl-Rodl’s Theorem and Variations on the Cap Set Problem: A Recent Research Project with Roy Meshulam (A)

Voita Rodl

I would like to tell you about a research project in progress with Roy Meshulam. (We started it in the summer, but then moved to other things;  so far there are interesting insights, and perhaps problems, but not substantial results. Noga Alon contributed an important observation, and Jeff Kahn and Gabor Simonyi helpful comments.)

Roy and I are friends and have been discussing various mathematical problems for decades. In the last few years we have studied together topological Helly-type theorems and have written several papers together. Our most recent success was finding a topological version of a Helly-type theorem by Nina Amenta. This post is on another problem.

As always, and more so in the spirit of Nielsen’s and Gowers’s recent suggestions:  comments, ideas, related problems, lemmas, and discussions  are welcome.  Links or references to earlier works (published and unpublished) are also welcome.  I thought to wait with this story after I discuss the Frankl-Wilson theorem and the Frankl-Rodl theorem in my extremal combinatorics series. But as this project is somewhat related to the density Hales-Jewett project discussed over Gowers’s blog (and even more to this thread at Tao’s blog), I decided to go ahead with it.

## Part A: Finding generalized solutions for the cap set problem

### 1. Modular-lines-free-subsets

Let $n=3m$. A modular line in $T(n)= \{0,1,2\}^n$ is a set of  three elements $x,y,z$ such that the number of coordinates $i$ for which $x_i + y_i + z_i =a$ is zero modulo $m$ for $a=0,1,2$. (The sum $x_i+y_i+z_i$ is taken modulo 3.)

Problem 1:  How large can a subset $F$ of $T(n)$ be without containing a modular line? Prove that $|F| < (3-\epsilon)^n$, for some constant $\epsilon >0$.

### 2. Why? avoiding lines (3-term arithmetic progressions) in $Z_3^n$..

The following is a well-known problem:

Problem 2: Find the largest size $f(n)$ of a set $A \subset Z_3^n$ without an arithmetic progression of size 3. In this case an arithmetic progression of size three is simply a sequence of three elements $x, y$ and $z$ such that $x+y+z=0$. In other words, an arithmetic progression is simply an affine line.

This problem was discussed by Frankl, Graham, and Rodl in 1987 (and perhaps was posed even earlier) and they showed that $f(n)=o(3^n)$. Roy Meshulam proved that $f(n) \le 3^n/n$. This is a beautiful and clear application of Roth’s argument (for his famous theorem on the density of sets without 3-term arithmetic progressions,) for the case of subsets of $\{1,2,\dots,n\}$. Ten years ago Roy and I tried quite hard to push this Fourier-theoretic argument and to get a better bound, even slightly, even very slightly. We did not succeed. (And so far, nobody else has; a recent impressive result of Tom Sanders for 3-term arithmetic progressions without a repeated element for subsets of  $Z_4^n$ gives reason for optimism.) Let us write $f(n)=3^n/g(n)$. The best lower bound on $g(n)$ is $n$. There is a construction of Edell showing that $f(n) \ge2.2^n$ and thus the best upper bound for $g(n)$ is more than $1.3^n$. The gap is huge! What is the true value? Terry Tao wrote a lovely post on this problem. A set in $Z_3^n$ without a 3-term arithmetic progression (or alternatively not containing an affine line) is called a cap set.

Let me mention that the set of 0-1 vectors of length $n$ is a very simple example of a cap set of size $2^n$; choosing a set at random works if the size is less than $3^{n/3}$.   Problem 1 is a variation on problem 2.  Continue reading