# Difference between revisions of "The Erdos-Rado sunflower lemma"

## The problem

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 one of the sets belongs to all of them. A basic and simple result of Erdos and Rado asserts that

Erdos-Rado Delta-system 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 $r$.

(We denote by $f(k,r)$ the smallest integer that suffices for the assertion of the theorem to be true.) The simple proof giving $f(k,r)\le k! (r-1)^k$ can be found here.

The best known general upper bound on $f(k,r)$ (in the regime where $r$ is bounded and $k$ is large) is

$\displaystyle f(k,r) \leq D(r,\alpha) k! \left( \frac{(\log\log\log k)^2}{\alpha \log\log k} \right)^k$

for any $\alpha \lt 1$, and some $D(r,\alpha)$ depending on $r,\alpha$, proven by Kostkocha from 1996. The objective of this project is to improve this bound, ideally to obtain the Erdos-Rado conjecture

$\displaystyle f(k,r) \leq C^k$

for some $C=C(r)$ depending on $r$ only. This is known for $r=1,2$ but remains open for larger r.