# A lecture by Noga

Noga with Uri Feige among various other heroes

A few weeks ago I devoted a post to the 240-summit conference for Péter Frankl, Zoltán Füredi, Ervin Győri and János Pach, and today I will bring you the slides of Noga Alon’s lecture in the meeting. Noga is my genious twin academic brother – we both were graduate students under the supervision of Micha A. Perles in the same years and we both went to MIT as postocs in fall 1983.  The lecture starts with briefly mentioning four results by the birthday boys related to combinatorics and geometry and continues with recent startling results by Alon, Ankur Moitra, and Benny Sudakov. One out of many contributions of Noga over the years is building a large infrastructure of constructions and examples, often very surprising,  in combinatorics, graph theory, information theory,TOC, and related areas. And the new results add to this infrastructure. The slides are very clear. Enjoy!

# My Mathematical Dialogue with Jürgen Eckhoff

Jürgen Eckhoff, Ascona 1999

Jürgen Eckhoff is a German mathematician working in the areas of convexity and combinatorics. Our mathematical paths have met a remarkable number of times. We also met quite a few times in person since our first meeting in Oberwolfach in 1982. Here is a description of my mathematical dialogue with Jürgen Eckhoff:

Summary 1) (1980) we found independently two proofs for the same conjecture; 2) (1982) I solved Eckhoff’s Conjecture; 3) Jurgen (1988) solved my conjecture; 4) We made the same conjecture (around 1990) that Andy Frohmader solved in 2007,  and finally  5) (Around 2007) We both found (I with Roy Meshulam, and Jürgen with Klaus Peter Nischke) extensions to Amenta’s Helly type theorems that both imply a topological version.

(A 2009 KTH lecture based on this post or vice versa is announced here.)

Let me start from the end:

### 5. 2007 – Eckhoff and I  both find related extensions to Amenta’s theorem.

Nina Amenta

Nina Amenta proved a remarkable extension of Helly’s theorem. Let $\cal F$ be a finite family with the following property:

(a) Every member of $\cal F$ is the union of at most r pairwise disjoint compact convex sets.

(b) So is every intersection of members of $\cal F$.

(c) $|{\cal F}| > r(d+1)$.

If every r(d+1) members of $\cal F$ has a point in common, then all members of $\cal F$ have a point in common!

The case r=1 is Helly’s theorem, Grünbaum and Motzkin proposed this theorem as a conjecture and proved the case r=2. David Larman  proved the case r=3.

Roy Meshulam

Roy Meshulam and I studied a topological version of the theorem, namely you assume that every member of F is the union of at most r pairwise disjoint contractible compact sets in $R^d$ and that all these sets together form a good cover – every nonempty intersection is either empty or contractible. And we were able to prove it!

Eckhoff and Klaus Peter Nischke looked for a purely combinatorial version of Amenta’s theorem which is given by the old proofs (for r=2,3) but not by Amenta’s proof. An approach towards such a proof was already proposed by Morris in 1968, but it was not clear how to complete Morris’s work. Eckhoff and Nischke were able to do it! And this also implied the topological version for good covers.

The full results of Eckhoff and Nischke and of Roy and me are independent. Roy and I showed that if the nerve of $\cal G$ is d-Leray then the nerve of $\cal F$ is ((d+1)r-1)-Leray. Eckhoff and showed that if the nerve of $\cal G$ has Helly number d, then the nerve of $\cal F$ has Helly number (d+1)r-1. Amenta’s argument can be used to show that if the nerve of $\cal G$ is d-collapsible then the nerve of F is  ((d+1)r-1)-collapsible.

Here, a simplicial comples K is d-Leray if all homology groups $H_i(L)$ vanishes for every $i \ge d$ and every induced subcomplex L of K.

Roy and I were thinking about a common homological generalization which will include both results but so far could not prove it.

# 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.

# Test Your Intuition (14): A Discrete Transmission Problem

Recall that the $n$-dimensional discrete cube is the set of all binary vectors ($0-1$ vectors) of length n. We say that two binary vectors are adjacent if they differ in precisely one coordinate. (In other words, their Hamming distance is 1.) This gives the $n$-dimensional discrete cube a structure of a graph, so from now on , we will refer to binary vectors as vertices.

Suppose that you want to send a message so that it will reach all vertices of the discrete $n$-dimensional cube. At each time unit (or round) you can send the message to one vertex. When a vertex gets the message at round $i$ all its neighbors will receive it at round $i+1$.

The question is

how many rounds it will take to transmit the message to all vertices?

Here is a very simple way to go about it: At round 1 you send the message to vertex (0,0,0,…,0). At round 2 you send the message to vertex (1,1,…,1). Then you wait and sit. With this strategy it will take $\lceil n/2\rceil+1$ rounds.

Test your intuition: Is there a better strategy?