Improved Bounds for the Sunflower Lemma Ryan Alweiss∗ Shachar Lovett† Princeton University University of California, San Diego United States United States [email protected] [email protected] Kewen Wu Jiapeng Zhang‡ Peking University Harvard University China United States [email protected] [email protected] ABSTRACT Defnition 1.1 (Sunfower). A collection of sets S1;:::; Sr is an if A sunfower with r petals is a collection of r sets so that the inter- r-sunfower section of each pair is equal to the intersection of all. Erdős and Si \ Sj = S1 \···\ Sr ; 8i , j: Rado proved the sunfower lemma: for any fxed r, any family of We call = 1 \···\ the and 1 n n the sets of size , with at least about w sets, must contain a sunfower. K S Sr kernel S K;:::; Sr K petals w w of the sunfower. The famous sunfower conjecture is that the bound on the number of sets can be improved to cw for some constant c. In this paper, we Erdős and Rado [13] proved that large enough set systems must improve the bound to about ¹logwºw . In fact, we prove the result contain a sunfower. The name “sunfower" is due to Peter Frankl. for a robust notion of sunfowers, for which the bound we obtain is Lemma 1.2 (Sunflower lemma [13]). Let r ≥ 3 and F be a w-set tight up to lower order terms. system of size jF j ≥ w! · ¹r − 1ºw . Then F contains an r-sunfower. CCS CONCEPTS Erdős and Rado conjectured in the same paper that the bound in Lemma 1.2 can be drastically improved. • Mathematics of computing ! Combinatorics; • Theory of computation ! Pseudorandomness and derandomization; Com- Conjecture 1.3 (Sunflower conjecture [13]). Let r ≥ 3. There plexity theory and logic. exists c¹rº such that any w-set system F of size jF j ≥ c¹rºw contains an r-sunfower. 1+ 1 KEYWORDS The bound in Lemma 1.2 is of the form ww¹ o¹ º where the o¹1º Robust sunfower lemma, sunfower conjecture, switching lemma depends on r. Despite nearly 60 years of research, the best known bounds towards the sunfower conjecture were still of the form ACM Reference Format: 1+ 1 Ryan Alweiss, Shachar Lovett, Kewen Wu, and Jiapeng Zhang. 2020. Im- ww¹ o¹ ºº, even for r = 3. More precisely, Kostochka [27] proved proved Bounds for the Sunfower Lemma. In Proceedings of the 52nd An- that any w-set system of size jF j ≥ cw! · ¹log log logw/log logwºw nual ACM SIGACT Symposium on Theory of Computing (STOC ’20), June must contain a 3-sunfower for some absolute constant c. Recently, 3 4+ 1 22–26, 2020, Chicago, IL, USA. ACM, New York, NY, USA, 7 pages. https: Fukuyama [17] claimed an improved bound for r = 3 tow¹ / o¹ ººw , //doi.org/10.1145/3357713.3384234 but this proof has yet to be verifed. In this paper, we vastly improve the known bounds. We prove 1+ 1 1 INTRODUCTION that any w-set system of size ¹logwºw¹ o¹ ºº must contain a sun- Let X be a fnite set. A set system F on X is a collection of subsets fower. More precisely, we obtain the following: of . We call a -set system if each set in has size at most . X F w F w Theorem 1.4 (Main theorem, sunflowers). Let r ≥ 3. Any w-set system F of size jF j ≥ ¹logwºw ¹r · log logwºO¹wº contains ∗Research supported by an NSF Graduate Research Fellowship. †Research supported by NSF award 1614023. an r-sunfower. ‡Research Supported by NSF grant CCF-1763299 and Salil Vadhan’s Simons Investiga- tor Award. 1.1 Robust Sunfowers We consider a “robust" generalization of sunfowers, the study Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed of which was initiated by Rossman [40], who was motivated by for proft or commercial advantage and that copies bear this notice and the full citation questions in complexity theory. Later, it was studied by Lovett, on the frst page. Copyrights for components of this work owned by others than the Solomon and Zhang [29] in the context of the sunfower conjecture. author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specifc permission First, we defne a more “robust" version of the property of hav- and/or a fee. Request permissions from [email protected]. ing disjoint sets. Given a fnite set X, we denote by U¹X;pº the STOC ’20, June 22–26, 2020, Chicago, IL, USA distribution of sets Y ⊂ X, where each element x 2 X is included in © 2020 Copyright held by the owner/author(s). Publication rights licensed to ACM. ACM ISBN 978-1-4503-6979-4/20/06...$15.00 Y independently with probability p (there are sometimes referred https://doi.org/10.1145/3357713.3384234 to as “p-biased distributions”). 624 STOC ’20, June 22–26, 2020, Chicago, IL, USA Ryan Alweiss, Shachar Lovet, Kewen Wu, and Jiapeng Zhang Defnition 1.5 (Satisfying set system). Let 0 < α; β < 1. A set 1.2 Connections to Computer Science system F on X is ¹α; βº-satisfying if The sunfower lemma has had many applications in mathematics and computer science. Here we briefy discuss some of the computer Pr »9S 2 F ; S ⊂ Y ¼ > 1 − β: Y ∼U¹X;αº science applications. While it is reasonable to assume that some of the bounds obtained using the sunfower lemma can be improved As aforementioned, the property of being satisfying is a robust using our new results, we have not attempted a thorough literature analogue of the property of having disjoint sets. survey to see which ones can be improved, and leave this for future work. Lemma 1.6 ([29]). If F is a ¹1/r; 1/rº-satisfying set system and ; < F , then F contains r pairwise disjoint sets. Circuit lower bounds. Alon, Karchmer, and Wigderson [1] used the sunfower lemma to prove a lower bound for the number of Proof. Let be a set system on . Consider a random -coloring F X r wires in a circuit that computes the Hadamard transform by depth- of , where each element obtains any of the colors with equal X r 2 circuits. Jukna [25] extended this work, and also used the sun- probability. Let denote the color classes, which are a Y1;:::;Yr fower lemma to prove similar lower bounds for approximating the random partition of . For = 1 , let denote the event that X i ;:::;r Ei Hadamard transform. As aforementioned, Rossman [40] defned contains an -monochromatic set, namely, F i robust sunfowers, motivated by an application to monotone cir- cuit lower bounds. The improved (robust) sunfower lemma has Ei = »9S 2 F ; S ⊂ Yi ¼ : been used by Cavalar, Kumar, and Rossman [6] to improve previous Note that Yi ∼ U¹X; 1/rº, and since we assume F is ¹1/r; 1/rº- monotone circuit lower bounds. satisfying, we have Dinur and Safra [11] used sunfowers Pr»E ¼ > 1 − 1/r: Hardness of approximation. i in the soundness analysis of their proof of hardness of approxima- By the union bound, with positive probability all E1;:::; Er hold. tion for the Minimum Vertex Cover problem. In this case, F contains a set which is i-monochromatic for each i = 1;:::;r. Such sets must be pairwise disjoint. □ Matrix multiplication. Alon et al. [2] studied variants of the sun- fower conjecture and their connections with fast matrix multipli- Given a set system F on X and a set T ⊂ X, the link of F at T is cation algorithms. FT = fS n T : S 2 F ;T ⊂ Sg: Pseudorandomness. Gopalan, Meka, and Reingold [20] used the robust sunfowers for DNF sparsifcation, which gives better pseu- We now formally defne a robust sunfower (which was called a dorandom generators fooling small-width DNFs and faster deter- quasi-sunfower in [40] and an approximate sunfower in [29]). ministic algorithm counting satisfying assignments of DNFs. Let 0 1, be a set Defnition 1.7 (Robust sunfower). < α; β < F Luby et al.[32] studied broadcast encryption sys- system, and let = Ñ be the common intersection of all sets Cryptography. K S 2F S tem, and proved a trade-of between the number of establishment in . is an -robust sunfower if (i) < ; and (ii) is F F ¹α; βº K F FK keys held by each user and the number of transmissions needed -satisfying. We call the . ¹α; βº K kernel to establish a new broadcast key, where their lower bounds relies Lemma 1.8 ([29]). Any ¹1/r; 1/rº-robust sunfower contains an on the sunfower lemma. Naor et al.[35] extended this to a wider r-sunfower. regime of parameters. Gentry et al.[19] proved that the bounds are optimal using a weaker notion of sunfowers. Dachman-Soled et al. [8] studied locally decodable and updatable Proof. Let F be a ¹1/r; 1/rº-robust sunfower, and let K = Ñ be the common intersection of the sets in . Note that non-malleable codes. They showed that a sunfower structure in S 2F S F by assumption, F does not contain the empty set as an element. the codewords allows for a rewind attack. K Komargodski et al. [26] showed that fnding a sunfower (or a Lemma 1.6 gives that F contains r pairwise disjoint sets S1;:::; Sr .