The Erdos-Rado sunflower lemma
The problem
A sunflower (a.k.a. Delta-system) of size [math]\displaystyle{ r }[/math] is a family of sets [math]\displaystyle{ A_1, A_2, \dots, A_r }[/math] 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 [math]\displaystyle{ f(k,r) }[/math] so that every family [math]\displaystyle{ \cal F }[/math] of [math]\displaystyle{ k }[/math]-sets with more than [math]\displaystyle{ f(k,r) }[/math] members contains a sunflower of size [math]\displaystyle{ r }[/math].
(We denote by [math]\displaystyle{ f(k,r) }[/math] the smallest integer that suffices for the assertion of the theorem to be true.) The simple proof giving [math]\displaystyle{ f(k,r)\le k! (r-1)^k }[/math] can be found here.
The best known general upper bound on [math]\displaystyle{ f(k,r) }[/math] (in the regime where [math]\displaystyle{ r }[/math] is bounded and [math]\displaystyle{ k }[/math] is large) is
- [math]\displaystyle{ \displaystyle f(k,r) \leq D(r,\alpha) k! \left( \frac{(\log\log\log k)^2}{\alpha \log\log k} \right)^k }[/math]
for any [math]\displaystyle{ \alpha \lt 1 }[/math], and some [math]\displaystyle{ D(r,\alpha) }[/math] depending on [math]\displaystyle{ r,\alpha }[/math], proven by Kostkocha from 1996. The objective of this project is to improve this bound, ideally to obtain the Erdos-Rado conjecture
- [math]\displaystyle{ \displaystyle f(k,r) \leq C(r)^k }[/math]
for some [math]\displaystyle{ C(r) }[/math] depending on [math]\displaystyle{ k }[/math]. This is known for [math]\displaystyle{ r=1,2 }[/math] but remains open for larger r.
Threads
- Polymath10: The Erdos Rado Delta System Conjecture, Gil Kalai, Nov 2, 2015. Active
Wikipedia links
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
- The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets, R. Ahlswede, L. Khachatrian, Journal of Combinatorial Theory, Series A 76, 121-�138 (1996).
- 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.
- Extremal problems on [math]\displaystyle{ \Delta }[/math]-systems, A. V. Kostochka
- On Erdos' extremal problem on matchings in hypergraphs, T. Luczak, K. Mieczkowska
- Intersection theorems for systems of sets, J. H. Spencer, Canad. Math. Bull. 20 (1977), 249-254.
