The purpose of this post is to start the polymath10 project. It is one of the nine projects (project 3d) proposed by Tim Gowers in his post “possible future polymath projects”. The plan is to attack Erdos-Rado delta system conjecture also known as the sunflower conjecture. We will start the research thread here. Mimicking a feature of polymath1 I will propose a detailed approach in the next post. Here, I will remind you of the conjecture and some basic known results, mention a few observation and ingredients of my point of view, and leave the floor to you comments.

The Erdos-Rado Delta-system Theorem and Conjecture

A sunflower (a.k.a. Delta-system) of size is a family of sets such that every element that belongs to more than one of the sets belongs to all of them. A basic and simple result of Erdos and Rado asserts that

Erdos-Rado Delta-syatem theorem: There is a function so that every family of -sets with more than members contains a sunflower of size .

(We denote by the smallest integer that suffices for the assertion of the theorem to be true.) The simple proof giving can be found here.

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

The Erdos-Rado conjecture: Prove that .

Here, is a constant depending on . It may be also the case that we can take for some absolute constant C. The conjecture is already most interesting for . And getting progress for will already be great.

At least for the delta system conjecture is true. Every two sets form a Delta-system of size two. So . Can we find more difficult proofs and for weaker statements? The case will play a role in the context of more general questions.

The best known upper bounds

An excellent review paper is Extremal problem on Δ-systems by Alexandr Kostochka. After an early paper by L. Abbott, D. Hanson, and N. Sauer imroving both the upper and lower bounds, Joel Spencer proved an upper bound of for every fixed . Spencer also proved an upper bound for . (The exponent was improved further to 1/2 by Furedi and Kahn.) A remarkable result by Sasha Kostochka from 1996 is the best upper bound known today.

Sasha Kostochka

A summary of my proposed approach:

I. A more general problem

Given a family of sets and a set , the star of is the subfamily of those sets in containing , and the link of is obtained from the star of by deleting the elements of from every set in the star. (We use the terms link and star because we do want to consider eventually hypergraphs as geometric/topological objects.)

We can restate the delta system problem as follows: f(k,r) is the maximum size of a family of k-sets such that the link of every set A does not contain r pairwise disjoint sets.

What can we say about families of k-sets from {1,2,…,n} such that that the link of every set A of size at most m-1 does not contain r pairwise disjoint sets? In particular, what is f(k,r,m;n) the maximum number of sets in such a family. We can ask

Question 1: Understand the function f(k,r,m;n).

Question 2: Is it true that , where is a constant depending on r, perhaps even linear in r.

My proposal is to approach the Delta-system problem via these questions. We note that Question 1 includes the Erdos-Ko-Rado theorem:

(r=2, m=1) Erdös-Ko-Rado Theorem: An intersecting of k-subsets of , when contains at most sets.

Here, a family of sets is “intersecting” if every two sets in the family has non empty intersection. The situation for and general was raised by Erdos-Ko-Rado (who proposed a conjecture for a certain special case), Frankl proposed a general conjecture that was settled by Alswede-Khachatrian. This was a remarkable breakthrough. The case of general r and m=1 is again a famous question of Erdos. The conjecture is that when , This is a classic result by Erdos and Gallai (1959) for graphs (k=2), and very recently it was proved for r=3 for large values of , in the paper On Erdos’ extremal problem on matchings in hypergraphs by Tomasz Luczak, and Katarzyna Mieczkowska, and for all values of by Peter Frankl.

We want much weaker results (suggested by Problem 2) than those given (or conjectured) by Erdos-Ko-Rado theory, but strong enough to apply to the Delta system conjecture.

II. moving to the multipartite case

A family of -sets is balanced (or -colored) if it is possible to color the elements with colors so that every set in the family is colorful.

Reduction (folklore): It is enough to prove Erdos-Rado Delta-system conjecture for the balanced case.

Proof: Divide the elements into d color classes at random and take only colorful sets. The expected size of the surviving colorful sets is .

III: Enters homology

A family of -sets is acyclic (with Z2 coefficient) if it contains no -cycle. A -cycle is a family of -sets such that every set of size is included in an even number of -sets in .

Theorem 1: An acyclic family of k-subsets of [n], contains at most sets.

I suppose many people ask themselves:

Question 3: Are there some connections between the property “intersecting” and the property “acyclic?”

Unfortunately, but not surprisingly intersecting families are not always acyclic. And acyclic families are not always intersecting.(The condition from EKR theorem also disappeared in the result about acyclic families.)

Lets ignore for a minute that being acyclic and being intersecting are not related and ask.

Question 4: Is there some “acyclicity” condition related to (or analogous to) the property of not having 3 pairwise disjoint sets? r pairwise disjoint sets?

We know a few things about it.

Question 5: What can we say about families which are acyclic and so are all links for every set A of size at most m-1?

Here under some additional conditions there are quite a lot we can say in a direction of bounds asked for in Question 2.

(We note that one place where a connection between homology and Erdos-Ko-Rado theory was explored successfully is in the paper Homological approaches to two problems on finite sets by Rita Csákány and Jeff Kahn.)

IV) Coloring to the rescue!

Relating acyclicity and being intersecting is not easy in spite of the similar upper bound. We can ask now if for balanced families, there are some connections between the property “intersecting” and the property “acyclic?”

Question 6: Let be a balanced intersecting family of -sets, is acyclic?

The answer is yes. Eran Nevo pointed out a simple inductive argument which also extends in various directions. If you have a balanced k-dimensional cycle (mod Z/Z2) then by induction the link of a vertex v (which is also a cycle) has two disjoint sets R and R‘ and taking one of those sets with v and the other set with yet another vertex w yield a disjoint pair. (Each set of size k-1 in a cycle must be included in more than one sets of size k; in fact this is the only fact we are using.)

On technical matters: The project will run over this blog and Karim Adiprasito will join me in organizing it. (Perhaps to make the mathematical formulas appearing better we will move to another appearance.)