# 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 $r=1,2$(indeed we have $f(k,r)=1$ in those cases) but remains open for larger r.

## Variants and notation

Given a family F of sets and a set S, the star of S is the subfamily of those sets in F containing S, and the link of S is obtained from the star of S by deleting the elements of S 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.

Let f(k,r;m,n) denote the largest cardinality of a family 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. Thus f(k,r) = f(k,r;k,n) for n large enough.

Conjecture 1: $f(k,r;m,n) \leq C_r^k n^{k-m}$ for some $C_r$ depending only on r.

This conjecture implies the Erdos-Ko-Rado conjecture (set m=k). The Erdos-Ko-Rado theorem asserts that

$f(k,2;1,n) = \binom{n-1}{k-1}$ (1)

when $n \geq 2k$, which is consistent with Conjecture 1. More generally, Erdos, Ko, and Rado showed

$f(k,2;m,n) = \binom{n-m}{k-m}$

when $n$ is sufficiently large depending on k,m. The case of smaller n was treated by several authors culminating in the work of Ahlswede and Khachatrian.

Erdos conjectured that

$f(k,r;1,n) = \max( \binom{rk-1}{k}, \binom{n}{k} - \binom{n-r}{k} )$

for $n \geq rk$, generalising (1). This was established for k=2 by Erdos and Gallai, and for r=3 by Frankl (building on work by Luczak-Mieczkowska).

A family of k-sets is balanced (or k-colored) if it is possible to color the elements with k 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.

<B?Proof</B>: Divide the elements into d color classes at random and take only colorful sets. The expected size of the surviving colorful sets is k!/k^k|F|.

Hyperoptimistic conjecture: The maximum size of a balanced collection of k-sets without a sunflower of size r is (r-1)^k.

Disproven for k=3,r=3: set $|V_1|=|V_2|=|V_3|=3\ltmath\gt and use ijk to denote the 3-set consisting of the i^th element of V_1, j^th element of V_2, and k^th element of V_3. Then 000, 001, 010, 011, 100, 101, 112, 122, 212 is a balanced family of 9 3-sets without a 3-sunflower. == Threads == * [https://gilkalai.wordpress.com/2015/11/03/polymath10-the-erdos-rado-delta-system-conjecture Polymath10: The Erdos Rado Delta System Conjecture], Gil Kalai, Nov 2, 2015. \ltB\gtActive\lt/B\gt == External links == * [https://en.wikipedia.org/wiki/Sunflower_(mathematics) Sunflower (mathematics)] (Wikipedia article) * [http://mathoverflow.net/questions/163689/what-is-the-best-lower-bound-for-3-sunflowers What is the best lower bound for 3-sunflowers?] (Mathoverflow) == Bibliography == Edits to improve the bibliography (by adding more links, Mathscinet numbers, bibliographic info, etc.) are welcome! * Intersection theorems for systems of sets, H. L. Abbott, D. Hanson, and N. Sauer, J. Comb. Th. Ser. A 12 (1972), 381–389 * [https://www.math.uni-bielefeld.de/ahlswede/homepage/public/114.pdf The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets], R. Ahlswede, L. Khachatrian, Journal of Combinatorial Theory, Series A 76, 121-138 (1996). * [http://renyi.hu/~p_erdos/1960-04.pdf Intersection theorems for systems of sets], P. Erdős, R. Rado, Journal of the London Mathematical Society, Second Series 35 (1960), 85–90. * [http://arxiv.org/abs/1205.6847 On the Maximum Number of Edges in a Hypergraph with Given Matching Number], P. Frankl * An intersection theorem for systems of sets, A. V. Kostochka, Random Structures and Algorithms, 9(1996), 213-221. * [http://www.math.uiuc.edu/~kostochk/docs/2000/survey3.pdf Extremal problems on \ltmath\gt\Delta$-systems], A. V. Kostochka