Improved bounds for the sunflower lemma

Abstract

A sunflower with r petals is a collection of r sets so that the intersection of each pair is equal to the intersection of all. Erdős and Rado proved the sunflower lemma: for any fixed r, any family of sets of size w, with at least about w w sets, must contain a sunflower. The famous sunflower conjecture is that the bound on the number of sets can be improved to c w for some constant c. In this paper, we improve the bound to about (log w) w . In fact, we prove the result for a robust notion of sunflowers, for which the bound we obtain is tight up to lower order terms.

Set system intersections can typically be blown up

“…Somewhat recently, the author, Lovett, Wu, and Zhang [1] improved the best known bounds for the Erdős-Rado sunflower lemma [4]. Central to this result is an "encoding" argument, in which the ground set X of a set system F is colored in some way.…”

Unavoidable hypergraphs

“…Despite significant efforts, a solution to this conjecture remains elusive. The current record is f r (k) ≤ O(k log(kr)) r , established in 2019 by Rao [38], building upon a breakthrough of Alweiss, Lovett, Wu and Zhang [2]. Some 43 years ago, Duke and Erdős [11] initiated the systematic investigation of a closely related problem.…”

“…where in the last inequality we used the assumption on t. We delete from H any edge containing two vertices in X. We deleted at most (2). Hence, vertices in X after deletion still have degree at least δ 2 .…”

“…At this point we have used at most d 2 vertices from C and kd 2 + d ≤ 2kd 2 vertices from D (the first term being the contribution of D \ X and the second of X). In the link graph L x (note that this graph consists of edges with one vertex in C and one in D), already used vertices from C touch at most d 2 • n ≤ δ/8 edges in total, using (2) The final lemma for the 3-uniform case is the following. It is similar in spirit to the above one, except that it works with smaller sets and only finds a single star.…”

“…2 and |C| ≤ 4e/L 4 = 16td 2 k. This implies |A ∪ B| ≤ 32td 2 and |A ∪ B ∪ C| ≤ 48td 2 k. Hence the total number of edges with at least 2 vertices in A, or at least 3 vertices in A ∪ B, or all 4 vertices in A ∪ B ∪ C is upper bounded by t 2 d 2 n 2 + 2 10 t 3 d 6 n + 219 t 4 d 8 k 4 ≤ 2 6 d 5/2 n 2 + 2 10 k 3 d 6 n + 2 19 d 9 k 4 ≤ 2 6 n 3 k + 2 10 n 3 k + 2 19 n 3 k ≤ 2 20 kn 3 ≤ e/16,where we usedt 4 ≤ d and t ≤ k in the second inequality and n = kd 3 in the third. In particular, there are at least e/16 remaining edges (which do not touch used vertices) such that they have at least 1 vertex in D, at least 2 vertices in C ∪ D and at least 3 vertices in B ∪ C ∪ D. Furthermore, there exists a subset F of these edges of size |F | ≥ (e/16)/35 kn 3 which all have the same number of vertices in each of A, B, C and D, since there are 4 4+3 3 = 35 different types of edges according to how many vertices they have in each of the sets A, B, C and D. We distinguish several cases depending on how many vertices our edges in F have in D. Case (i).…”