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 , ∀i , j. Rado proved the sunfower lemma: for any fxed r, any family of We call = 1 ∩ · · · ∩ the and 1 \ \ 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 systems must improve the bound to about (logw)w . 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 |F | ≥ w! ·(r − 1)w . 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. • of computing → ; • Theory of computation → Pseudorandomness and derandomization; Com- Conjecture 1.3 (Sunflower conjecture [13]). Let r ≥ 3. There plexity theory and logic. exists c(r) such that any w-set system F of size |F | ≥ c(r)w contains an r-sunfower. 1+ 1 KEYWORDS The bound in Lemma 1.2 is of the form ww( o( ) where the o(1) 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- ww( o( )), 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 |F | ≥ cw! ·(log log logw/log logw)w 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 tow( / o( ))w , //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 (logw)w( o( )) must contain a sun- Let X be a fnite set. A set system F on X is a collection of 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 |F | ≥ (logw)w (r · log logw)O(w) 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 . Given a fnite set X, we denote by U(X,p) the STOC ’20, June 22–26, 2020, Chicago, IL, USA distribution of sets Y ⊂ X, where each element x ∈ 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 (α, β)-satisfying if The sunfower lemma has had many applications in mathematics and computer science. Here we briefy discuss some of the computer Pr [∃S ∈ F , S ⊂ Y ] > 1 − β. Y ∼U(X,α) 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 (1/r, 1/r)-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 = [∃S ∈ F , S ⊂ Yi ] . been used by Cavalar, Kumar, and Rossman [6] to improve previous Note that Yi ∼ U(X, 1/r), and since we assume F is (1/r, 1/r)- monotone circuit lower bounds. satisfying, we have Dinur and Safra [11] used sunfowers Pr[E ] > 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 = {S \ T : S ∈ F ,T ⊂ S}. 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 ∈F S tem, and proved a trade-of between the number of establishment in . is an -robust sunfower if (i) < ; and (ii) is F F (α, β) K F FK keys held by each user and the number of transmissions needed -satisfying. We call the . (α, β) K kernel to establish a new broadcast key, where their lower bounds relies Lemma 1.8 ([29]). Any (1/r, 1/r)-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 (1/r, 1/r)-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 ∈F S F by assumption, F does not contain the 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 . K pair of duplicate sets), where the underlying set system is given by Thus S1 ∪ K,..., Sr ∪ K is an r-sunfower in F . □ the output of a succinct circuit, is hard assuming the existence of The proof of Theorem 1.4 follows from the following stronger collision resistant hash functions. theorem, by setting = = 1 and applying Lemma 1.8. The α β /r In the bit probe model, Frandsen theorem verifes a conjecture raised in [29], and answers a question Data structure lower bound. et al. [15] used the sunfower lemma to prove lower bounds for of [40]. dynamic word problems, Mortensen et al. [34] used it to prove Theorem 1.9 (Main theorem, robust sunflowers). Let 0 < trade-ofs in two stage greater-than functions, and Rahman [37] α, β < 1. Any w-set system F of size |F | ≥ (logw)w ·(log logw · used it to analyze the increment operation for . log(1/β)/α)O(w) contains an (α, β)-robust sunfower. In cell probe model, Gal et al. [18] used the sunfower lemma to prove lower bounds for the redundancy/query time trade-of of 1+ 1 The bound of (logw)w( o( )) for robust sunfowers is sharp; it solutions to static data structure problems, and Natarajan et al. [36] 1 1 cannot be improved beyond (logw)w( −o( )). We give an example used a weaker notion of sunfowers to analyze non-adaptive data demonstrating this in Lemma 2.1. structures computing the minimum, median, and predecessor.

625 Improved Bounds for the Sunflower Lemma STOC ’20, June 22–26, 2020, Chicago, IL, USA

In the CRCW PRAM model, the sunfower lemma was used to compression for DNFs”. For more details on the connection to DNF prove lower bounds by Berkman et al. [5] and incomparability compression see [29–31]). results by Grolmusz et al. [21] in various models. What we show is that this conjecture is true with two modifca- tions: we are allowed to remove a small fraction of the sets in , Haviv et al. [23] showed that a refutation of F Property testing. and also remove a small random fraction of the elements in the base a variant of the sunfower conjecture implies a super-polynomial set . To be more precise, sample for = 1 log . lower bound on the query complexity of the canonical tester for X W ∼ U(X,p) p O( / w) We show that with high probability over , for most sets , cycle freeness. Balaji et al. [4] used the sunfower lemma to obtain W S ∈ F there exists a set ′ such that: (i) ′ ; and (ii) lower bounds on the query complexity of graph properties in the S ∈ F S \ W ⊂ S \ W ′ 2. Thus we can move to study the set system ′ node query setting. |S \ W | ≤ w/ F comprised of the S ′ \ W above, which “approximately covers” F . Fixed parameter complexity. The sunfower lemma is a common Note that F ′ is a (w/2)-set system which is κ′-bounded for κ′ ≈ κ. kernelization technique used in FPT algorithms [7, 12]. For example, In the actual proof, we will replace w/2 with (1 − ε)w for a small ε it was used for hitting set problems by Flum et al. [14], for constraint to improve the bounds. For details see Lemma 3.6. Applying this satisfaction problem under fnite Boolean constraint families by “reduction step” iteratively t = logw times reduces the size w dras- Marx et al. [33], for the subgraph test problem by Jansen et al. [24], tically, and then we can apply standard tools (Janson’s inequality, and for set matching problems by Dell et al. [9]. The last two works see Lemma 3.9). We get that if we sample W1,...,Wt ∼ U(X,p) also use it for graph packing problems. (formally, they are disjoint, but we ignore this detail here), then with good probability there exists ∈ F such that ⊂ 1 ∪ · · · ∪ . The technique developed in this S S W Wt Thresholds in random graphs. Setting · = 1/3 and the good probability to be 2/3 gives that F paper has been used by Frankston, Kahn, Narayanan, and Park [16] p t is (1/3, 1/3)-satisfying, as desired. to resolve a conjecture of Talagrand in random graph theory. To conclude, let us comment on how we prove the reduction step (Lemma 3.6). The main idea is to use an encoding lemma, 1.3 Proof Overview inspired by Razborov’s proof of Håstad’s switching lemma [22, 39]. In this section, we explain the high level ideas underlying the proof Concretely, for W ⊂ X and S ∈ F , we say that the pair (W , S) of Theorem 1.9. Our framework builds upon the work of Lovett, is bad if there is no S ′ ∈ F such that (i) S ′ \ W ⊂ S \ W ; and Solomon and Zhang [29]. Their main idea was to apply a structure (ii) |S ′ \ W | ≤ w/2. We show that bad pairs can be efciently vs. pseudo-randomness approach. However, the proof relied on encoded, crucially relying on the κ-boundedness condition. This a certain conjecture on the level of pseudo-randomness needed allows to show that for a random W it is very unlikely that there for the argument to go through. Our main technical result is a will be many bad sets. The (w/2)-set system F ′ is then taken to be resolution of this conjecture. F ′ = {S ′ \ W : S ′ ∈ F , |S ′ \ W | ≤ w/2}. To be concrete, we consider the problem of fnding a 3-sunfower, which corresponds in our framework to fnding a (1/3, 1/3)-robust 1.4 Other Related Works sunfower (see Lemma 1.8). Given 2, our goal is to fnd a w ≥ Other than the works already mentioned, there are two more related parameter = such that any -set system of size w must κ κ(w) w κ works that should be mentioned. The term “sunfower” was coined contain a 1 3 1 3 -robust sunfower, and hence also a 3-sunfower. ( / , / ) by Deza and Frankl [10], as the original term used by Erdős and Recall the defnition of links: = : . FT {S \ T S ∈ F ,T ⊂ S} Rado was -systems. After the current work was made available on We say that a -set system is -bounded if (i) w ; and (ii) ∆ w κ |F | ≥ κ the arXiv, Rao [38] simplifed the proof using information theory w−|T | for all non-empty (The actual defnition needed |FT | ≤ κ T techniques. in the proof is more delicate, see Defnition 3.1 for details). Let F be a w-set system of size |F | ≥ κw . Then either F is Paper organization. We give an example showing that our bounds w−|T | κ-bounded, or otherwise there is a link FT of size |FT | ≥ κ . for robust sunfowers are essentially tight in Section 2. We prove In the latter “structured" case, we can pass to the link and apply our main technical result, Theorem 1.9, in Section 3. induction. (This argument is implicit in the proof of Lemma 3.4.) Thus it sufces to consider the “pseudo-random" case of w-set 2 LOWER BOUND FOR ROBUST systems which are κ-bounded. In [28, 29], it was conjectured that SUNFLOWERS for some absolute , a log C -bounded is necessarily 1 3 1 3 - C ( w) F ( / , / ) In this section, we construct an example of a -set system without satisfying. We show that there is some = log 1+o(1) is sufcient w κ ( w) robust sunfower, even though it has size log w(1−o(1)). For con- (see Theorem 3.5), which represents our main technical contribution. ( w) creteness we fx = = 1 2, but the construction can be easily This completes the proof of Theorem 1.9. We also show that = α β / κ modifed for any other constant values of . We assume that log 1−o(1) is necessary (see Lemma 2.1), so this is tight. α, β w ( w) is large enough. We next explain how we obtain the bound on κ. Let F be a w-set √ system which is κ-bounded. In [29] it was conjectured that there Lemma 2.1. There exists a w-set system of size ((logw)/8)w− w = exists a 2 -set system ′ that “covers” ; for any set , 1 1 (w/ ) F F S ∈ F (logw)w( −o( )) which does not contain a (1/2, 1/2)-robust sunfower. there exists S ′ ∈ F ′ such that S ′ ⊂ S, and F ′ is κ′-bounded for ′ ′ κ ≈ κ. If this conjecture is true, then it is sufcient to prove that F Let c ≥ 1 be determined later. Let X1,..., Xw be pairwise disjoint is (1/3, 1/3)-satisfying, as this would imply that F is also (1/3, 1/3)- sets of size m = log(w/c), and let X be their union. Let Fb = X1 × satisfying (in the language of [29], this corresponds to “upper bound · · · × Xw be the w-set system containing all sets which contain

626 STOC ’20, June 22–26, 2020, Chicago, IL, USA Ryan Alweiss, Shachar Lovet, Kewen Wu, and Jiapeng Zhang exactly one element from each of the Xi . We frst argue that Fb is Defnition 3.2 (Bounded set system). Let s be a weight profle. A set not satisfying. system F is s-bounded if there exists a weight function σ : F 7→ Q+ such that (F , σ) is s-bounded. Claim 2.2. For c ≥ 1, Fb is not (1/2, 1/2)-satisfying. We note that one may always normalize a weight profle to have Proof. Let ∼ U( 1/2). We analyze the probability that some Y X, s0 = 1. However, keeping s0 as a free parameter helps to simplify Xi is disjoint from Y , which implies that no set in Fb is contained some of the arguments later. in . The probability is 1 1 2−m w = 1 1 w , which is Y − ( − ) − ( − c/w) The main idea is to show that set systems which are s-bounded, more than 1 2 for 1. □ / c ≥ for s appropriately chosen, are “random looking” and in particular must be (α, β)-satisfying. This motivates the following defnition. Unfortunately, Fb does contain a (1/2, 1/2)-robust sunfower with a large kernel. For example, if T contains exactly one element from Defnition 3.3 (Satisfying weight profle). Let 0 < α, β < 1.A each of X1,..., Xw−1, then FbT is isomorphic to Xw , and in partic- weight profle s is (α, β)-satisfying if any s-bounded set system is ular is (1/2, 1/2)-satisfying. (α, β)-satisfying. To overcome this, let 0 be determined later, and choose ε > The following lemma underlies our proof of Theorem 1.9. This to be a subsystem that satisfes: F ⊂ Fb is implicitly where the “induction” on the “structured part” of the |S ∩ S ′| ≤ (1 − ε)w, ∀S, S ′ ∈ F , S , S ′. set family is occurring. For example, we can obtain by a greedy procedure, each time F Lemma 3.4. Let 0 < α, β < 1 and w ≥ 2. Let κ = κ(w) > ′ ′ choosing an element S in Fb and deleting all S such that |S ∩ S | > 1 be a non-decreasing function of w such that the weight profle 1 . The number of such ′ is at most w  εw 2w εw . −1 −ℓ ( − ε)w S εw m ≤ m (1;κ ,...,κ ) is (α, β)-satisfying for all ℓ = 1,...,w. Then any 1 Hence we can obtain F of size |F | ≥ 2−wm( −ε)w . w-set system F of size |F | > κw must contain an (α, β)-robust sunfower. Claim 2.3. For c ≥ 1/ε, F does not contain a (1/2, 1/2)-robust sunfower. Proof. Assume a contradiction, and let F be a w-set system on of size w without an -robust sunfower. Choose Proof. Consider any set . We need to prove that does X |F | > κ (α, β) F K ⊂ X F w−|K | to minimize w. Let K ⊂ X be maximal so that |FK | > κ . Note not contain a (1/2, 1/2)-robust sunfower with kernel K. If it does, 0 that we cannot have |K | = w, as in this case |F | = 1 = κ , and so F must contain at least two sets, which implies that |K ∩ X | ≤ 1 K K i | | ≤ −1. Let F ′ = F \{∅}. Note that |F ′| ≥ w−|K |, where for for all i, and that in addition |K | ≤ (1 − ε)w. However, in this case K w K κ any non-empty setT disjoint from K, |F ′| = |F | ≤ κw−|K |−|T |. we claim even F is not (1/2, 1/2)-satisfying. T K∪T bK Let = 1 ′ for ′. Then ′ is 1; −1 −ℓ - To prove this, let I = {i : |K ∩ X | = 0} where |I | ≥ εw. Let σ(S) /|F | S ∈ F (F , σ) ( κ ,...,κ ) i bounded for = , so by minimality, ′ is -satisfying, ∼ U( 1/2). The probability that there exists ∈ such that ℓ w − |K | F (α, β) Y X, i I Y ′ −m |I | εw and hence {S ∪ K : S ∈ F } is an (α, β)-robust sunfower contained is disjoint from Xi is 1 − (1 − 2 ) ≥ 1 − (1 − c/w) which is in F . □ more than 1/2 for c ≥ 1/ε. □ To conclude the proof of Lemma 2.1 we optimize the parameters. Lemma 3.4 motivates the following defnition. For 0 < α, β < 1 √ and 2, let be the least such that 1; −1 −w Set = 1/ . We have |F | ≥ 2−w (log( ))(1−ε)w . Setting = 1/ w ≥ κ(w, α, β) κ ( κ ,...,κ ) c ε √ εw ε w 1 1 is (α, β)-satisfying. Theorem 1.9 follows by combining Lemma 3.4 gives |F | ≥ ((logw)/8)w− w = (logw)( −o( ))w . with the following theorem, which bounds κ(w, α, β). 1 3 PROOF OF THEOREM 1.9 Theorem 3.5. κ(w, α, β) ≤ logw ·(log logw · log(1/β)/α)O( ). We proceed to prove Theorem 1.9. The main idea is to apply a We note that Theorem 3.5 proves a conjecture raised in [29]. We structure vs. pseudo-randomness paradigm, following the approach prove Theorem 3.5 in the remainder of this section. outlined in [29]. Let F be a set system, and let σ : F 7→ Q≥0 be a weight function that assigns non-negative rational weights to sets 3.1 A Reduction Step in which are not all 0. For simplicity, we do not permit irrational F Let be a -set system on , and fx ′ . The main goal in weights. We call the pair a . For a F w X w ≤ w (F , σ) weighted set system this section is to reduce to a ′-set system ′. We prove the ′ we write ′ = Í the sum of the weights of F w F F ⊂ F σ(F ) S ∈F′ σ(S) following lemma in this section. the sets in F ′. A is a vector = ; where 1 ′ weight profle s (s0 s1,...,sk ) ≥ s0 ≥ Lemma 3.6. Let s = (s0;s1,...,sw ) be a weight profle, w ≤ w, 0 are rational numbers. We assume implicitly that ′ ′ ′ ′ s1 ≥ · · · ≥ sk ≥ δ > 0 and defne s = ((1 − δ)s0;s1,...,sw′ ). Assume s is (α , β )- si = 0 for all i > k. satisfying. Then for any p > 0, s is (α, β)-satisfying for Let be a weight (4/p)ws ′ Defnition 3.1 (Bounded weighted set system). s α = p + (1 − p)α ′, β = β ′ + w . profle that s = (s0;s1,...,sw ). A weighted set system (F , σ) is δs0 s-bounded if Let W ⊂ X. Given a set S ∈ F , the pair (W , S) is said to be good (i) σ(F ) ≥ s0; if there exists a set S ′ ∈ F (possibly with S ′ = S) such that (ii) for any link with non-empty . ′ σ(FT ) ≤ s |T | FT T (i) S \ W ⊂ S \ W . In particular, F is a w-set system. (ii) |S ′ \ W | ≤ w ′.

627 Improved Bounds for the Sunflower Lemma STOC ’20, June 22–26, 2020, Chicago, IL, USA

If no such S ′ exists, we say that (W , S) is bad. Note that ifW contains Proof of Lemma 3.6. Let (F , σ) be an s-bounded weighted set ′ ′ a set in F (i.e. S ⊂ W for some S ∈ F ) then all pairs (W , S) are system on X and s = (s0;s1,...,sw ). Let W ∼ U(X,p). Say that W good. is δ-bad if σ(B(W )) ≥ δs0. By applying Corollary 3.8 and Markov’s inequality, we obtain that Lemma 3.7. Let (F , σ) be an s = (s0;s1,...,sw )-bounded weigh -ted set system on X. Let W ⊂ X be a uniform subset of size |W | = E[σ(B(W ))] (4/p)ws ′ Pr[W is δ-bad] ≤ ≤ w . p|X | and B(W ) = {S ∈ F : (W , S) is bad}. Then EW [σ(B(W ))] ≤ δs0 δs0 (4/p)ws ′ . w Fix W which is not δ-bad. By assumption, if (W , S) is good for ′ ′ Proof. First, we simplify the setting a bit. We may assume by S ∈ F , then there exists π(S) = S ∈ F (possibly with S = S) such that (i) S ′ \ W ⊂ S \ W and (ii) |S ′ \ W | ≤ w ′. Choose such π with scaling σ and s by the same factor that σ(S) = NS , S ∈ F are all Í the smallest possible image so that if S ′, S ′′ in the image of π are integers. Let N = NS ≥ s0. We can identify (F , σ) with the multi- distinct then ′ \ , ′′ \ . set system F ′ = {S1,..., S }, where every set S ∈ F is repeated S W S W N Defne a new weighted set system ′ ′ on ′ = as times. Observe that |F ′| = (F ) and that ( ) is bad in F (F , σ ) X X \ W NS T σ T W , S ′ follows: if (W , Si ) is bad in F where Si = S is any copy of S. Thus ′ F ′ = { ( )\ : ∈ F \ B( )} ′( ′ \ ) = ( −1( ′)) σ(B(W )) = |{i : Si ∈ F and (W , Si ) is bad}|. π S W S W , σ S W σ π S . ′ We claim that ′ is ′ = 1 ; -bounded. To see Assume that (W , Si ) is bad in F . In particular, this means that F s (( − δ)s0 s1,...,sw′ ) that, note that ′ ′ = 1 and that for any W does not contain any set in F . We describe (W , Si ) with a small σ (F ) σ(F\B(W )) ≥ ( − δ)s0 ′ amount of information. Let |X | = n and |W | = pn. We encode set T ⊂ X , (W , Si ) as follows: ′ ′ Õ ′ ′ Õ Õ σ (F ) = σ (S ) = σ(S) ≤ σ(S) = σ(FT ) ≤ s | |. (1) The frst piece of information is ∪ . The number of T T W Si S′ ⊃T S:π (S)⊃T S ⊃T options for this is Finally, all sets in F ′ have size at most ′. Thus, if we choose w w Õ  n   n + w   n  ′ ′ ′ then we obtain that with probability more than ≤ ≤ p−w . W ∼ U(X , α ) pn + i pn + w pn 1 ′, there exist ∗ ′ such that ∗ ′. Recall that ∗ = i=0 −β S ∈ F S ⊂ W S S \W for some S ∈ F . Thus S ⊂ W ∪ W ′, which is distributed according (2) Given W ∪ S , let j be minimal such that S ⊂ W ∪ S ; in i j i to U(X,p + (1 − p)α ′). □ particular, this is equivalent to Sj \ W ⊂ Si \ W . There are fewer than 2w possibilities for A = S ∩S given that we know i j 3.2 A Final Step Sj . As such, we will let A be the second piece of information. ′ In this section, we directly show that bounded set systems (with (3) Note that as (W , Si ) is bad, |A| = |Sj \W | > w . So we know ′ very good bounds) are satisfying. A similar argument appears in a subset A of Si of size larger than w . The number of the ′ ′ [40]. sets in F which contain is |F | ≤ ′ . The third piece of A A sw information will be which one of these is . Si Lemma 3.9. 0 1 ≥ 2 = (2 + 4 ln(1/ )) · (4) Finally, once we have specifed , we will specify , Let < α, β < , w , and set κ β Si Si ∩ W = ; w w/α. Let (F , σ) be an s (s0 s1,...,sw )-bounded weighted set which is of course one of 2 possible subsets of Si . −i system where si < κ · s0. Then F is (α, β)-satisfying. From these four pieces of information one can uniquely recon- struct ( ). Thus the total number of bad pairs ( ) is bounded W , Si W , Si Proof. We can assume by scaling that NS = σ(S) for S ∈ F by are all integers. Let Fb be the multi-set system, where each S ∈ F     w −w n w w w n is repeated NS times and |F | > κ . Then we may also assume p · 2 · sw′ · 2 = (4/p) sw′ . pn pn that all sets in Fb have size exactly w, by adding diferent dummy n Í The number of sets W ⊂ X of size |W | = p|X | is . The lemma elements to each set of size below w. Let N = NS ≥ s0 and denote pn ′ = , where each is of width . Note that this ′ follows by taking expectation over W . □ F {S1,..., SN } Si w F satisfes the assumption of the lemma, and that for any set W ⊂ X, The following is a corollary of Lemma 3.7, where we replace if W contains a set of F ′ then it also contains a set of F . sampling W ⊂ X of size |W | = p|X | with sampling W ∼ U(X,p). The proof is by Janson’s inequality (see for example [3, Theorem 8.1.2]). Let W ∼ U(X, α) and Zi be the indicator variable for Si ⊂ Corollary 3.8. Let (F , σ) be an s = (s0;s1,...,s )-bounded w W . Denote i ∼ j if S , S intersect. Defne weighted set system on X. Let W ∼ U(X,p) and B(W ) = {S ∈ F : i j ( , ) } E [ (B( ))] ≤ (4/ )w ′ Õ Õ W S is bad . Then W σ W p sw . µ = E[Zi ], ∆ = E[Zi Zj ]. Proof. The proof is by a reduction to Lemma 3.7. Replace the i i∼j ′ w base set X with a much larger set X (without changing F , so the We have µ = Nα . To compute ∆, let pℓ denote the fraction of ′ ′ new elements do not belong to any set in F ). Let W ⊂ X be a pairs (i, j) such that |Si ∩ Sj | = ℓ. Then uniform set of size ′ = ′ , and let = ′ . Then as ′ |W | p|X | W W ∩ X X w gets bigger, the distribution of ′ approaches , while the Õ 2 2 W U(X,p) ∆ = p N α w−ℓ . conclusion of the lemma depends only on . □ ℓ W ℓ=1

628 STOC ’20, June 22–26, 2020, Chicago, IL, USA Ryan Alweiss, Shachar Lovet, Kewen Wu, and Jiapeng Zhang

−w ∗/2 To bound pℓ, note that for each Si ∈ F , and any R ⊂ Si of size then ∆r ≤ 2K . |R | Next, we apply Lemma 3.9 to bound 1 2 ∗ 2 and |R| = ℓ, the number of Sj ∈ F such that R ⊂ Sj is |FR | ≤ N /κ . A( / ,w ) ≤ α/ 1 ∗ Thus we can bound B(1/2,w∗) ≤ β/2. Observe that (1/2;κ− ,...,κ−w )-bounded set w w systems are also 1; 2 −1 2 −w ∗ -bounded, in which case Õ w 2 2 Õ  w ℓ 2 ( (κ/ ) ,..., (κ/ ) ) ∆ ≤ κ−ℓN α w−ℓ ≤ µ . we can apply Lemma 3.9 and obtain that we need =1 ℓ =1 ακ ℓ ℓ 1 + log 1 ∗ 2 κ ≥ Ω(( ( /β)) · w /α). Let κ = qw/α for q ≥ 2. Then ∆ ≤ 2µ /q. Note that in addition Let us now put the bounds together. We still have the freedom ∆ ≥ µ, as we include in particular the pairs (i,i) in ∆. Thus by Janson’s inequality, to choose ε > 0 and w∗ ≥ 1/ε. To obtain A(0,w) ≤ α, B(0,w) ≤ β, we also need 2 1 2. Thus all the constraints are:  2  ∆r ≤ β/ < / µ n q o ∗ Pr[∀i, Zi = 0] ≤ exp − ≤ exp − . (1) w ≥ 1/ε; 2∆ 4 1 (2) κ −ε = (K · u logw)/(εα) for some constant K ≥ 4; The lemma follows by setting = 2 + 4 ln 1 . □ q ( /β) (3) κ ≥ Ω((1 + log(1/β)) · w∗/α); ∗ 2 (4) 2K−w / ≤ β/2 ⇐= w∗ ≥ Ω(log(1/β)/log K). 3.3 Putting Everything Together Set ε = 1/log logw and w∗ = c · max {log logw, log(1/β)} for some We prove Theorem 3.5 in this subsection, where our goal is to bound c ≥ 1. Then we obtain that the result holds whenever ( ). We will apply Lemma 3.6 iteratively, until we decrease κ w, α, β ( 1+2/log log w enough to apply Lemma 3.9.  1  w κ = Ω max logw log logw, Fix w ≥ 2 throughout, and let κ > 1 to be optimized later. We frst α introduce some notation. For 0 1 1, defne weight profle < ∆ < , ℓ ≥ 1 2 = 1 ; −1 −ℓ . Let be bounds such (log(1/β)) , s(∆, ℓ) ( − ∆ κ ,...,κ ) A(∆, ℓ), B(∆, ℓ) α that any -bounded set system is -satisfying. s(∆, ℓ) (A(∆, ℓ), B(∆, ℓ)) 1 )! Lemma 3.6 applied to w ′ ≥ w ′′ and p, δ gives the bound log(1/β) log logw . α A(∆,w ′) ≤ A(∆ + δ,w ′′) + p, O(1) ′ In particular, it sufces to set κ = logw ·(log logw ·log(1/β)/α) . (4/ )w ′ + ′′ + p B(∆,w ) ≤ B(∆ δ,w ) ′′ . δ(1 − ∆)κw ACKNOWLEDGMENTS We apply this iteratively for some widths w0,...,wr . Set w0 = w We thank Noga Alon, Matthew Wales, Mohammadmahdi Jahanara, ∗ ∗ and wi+1 = ⌈(1 − ε)wi ⌉ for some ε as long as wi > w for some w . and anonymous reviewers for helpful suggestions on an earlier ∗ In particular, we need w ≥ 1/ε to ensure wi+1 < wi and we will version of this paper. We also thank Andrew Thomason, Gil Kalai, ∗ optimize ε,w later. The number of steps is thus r ≤ (u logw)/ε and Peter Frankl for pointing us to relevant references. for some constant u > 0. Let p1,...,pr and δ1,..., δr be the values we use for at each iteration. To simplify the notation, let = p, δ ∆i REFERENCES δ1 + ··· + δi and ∆0 = 0. 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