The Sunflower Conjecture

The Erdős–Rado Sunflower Theorem and Conjecture

A sunflower (also called a Delta-system) of size is a family of sets such that every element belonging to more than one of the sets belongs to all of them. Equivalently, every pairwise intersection is equal to the same common set (the core of the sunflower).

A basic and elegant result of Erdős and Rado asserts:

Erdős–Rado Delta-system theorem: There exists a function such that every family of -sets with more than members contains a sunflower of size .

(We denote by the smallest integer for which the theorem holds.) The simple proof giving the bound

can be found here.

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

The Erdős–Rado conjecture: Prove that

Here, is a constant depending only on . It may even be that one can take for some absolute constant .

The conjecture is already highly interesting for , and any substantial progress in this case would be a major breakthrough.

Polymath10 (2015)

Polymath10 was devoted to general discussions and ideas surrounding the sunflower problem, with particular emphasis on a homological/algebraic shifting approach. Here is the wiki page of the project, the first post, and the last post.

The breakthrough by Alweiss, Lovett, Wu, and Zhang

In 2019, a major breakthrough on the sunflower conjecture was achieved by Ryan Alweiss, Shachar Lovett, Kewen Wu, and Jiapeng Zhang.  Here is the paper. Anup Rao presented a simplified version here, and here are lecture notes by Shachar Lovett.

Their breakthrough gave the first substantial improvement over the classical Erdős–Rado bound in several decades. They proved that for every fixed ,

where is an absolute constant.

Thus, up to the extra factor, this comes strikingly close to the Erdős–Rado conjectured bound . Their proof introduced powerful new probabilistic and combinatorial ideas.