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Upper Bounds for Sunflower-Free Sets

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Upper Bounds for Sunflower-Free Sets Forum of Mathematics, Sigma (2017), Vol. 5, e15, 10 pages 1 doi:10.1017/fms.2017.12 UPPER BOUNDS FOR SUNFLOWER-FREE SETS ERIC NASLUND1 and WILL SAWIN2 1 Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA; email: [email protected] 2 ETH Institute for Theoretical Studies, ETH Zurich, 8092 Zurich,¨ Switzerland; email: [email protected] Received 16 July 2016; accepted 3 November 2016 Abstract A collection of k sets is said to form a k-sunflower, or ∆-system, if the intersection of any two sets from the collection is the same, and we call a family of sets F sunflower-free if it contains no 3-sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt (‘On n large subsets of Fq with no three-term arithmetic progression’, Ann. of Math. (2) 185 (2017), n 339–343); (‘Progression-free sets in Z4 are exponentially small’, Ann. of Math. (2) 185 (2017), 331–337) we apply the polynomial method directly to Erdos–Szemer˝ edi´ sunflower problem (Erdos˝ and Szemeredi,´ ‘Combinatorial properties of systems of sets’, J. Combin. Theory Ser. A 24 (1978), 308–313) and prove that any sunflower-free family F of subsets of f1; 2;:::; ng has size at most n.1Co.1// X n 3 jFj 3n : 6 k 6 22=3 k6n=3 We say that a set A ⊂ .Z=DZ/n D f1; 2;:::; Dgn for D > 2 is sunflower-free if for every distinct triple x; y; z 2 A there exists a coordinate i where exactly two of xi ; yi ; zi are equal. Using a version of the polynomial method with characters χ V Z=DZ ! C instead of polynomials, we show that any sunflower-free set A ⊂ .Z=DZ/n has size j j n A 6 cD D 3 − 2=3 where cD 22=3 .D 1/ . This can be seen as making further progress on a possible approach to proving the Erdos˝ and Rado sunflower conjecture (‘Intersection theorems for systems of sets’, J. Lond. Math. Soc. (2) 35 (1960), 85–90), which by the work of Alon et al. (‘On sunflowers and matrix multiplication’, Comput. Complexity 22 (2013), 219–243; Theorem 2.6) is equivalent to proving that cD 6 C for some constant C independent of D. 2010 Mathematics Subject Classification: 05D05 (primary); 05B40 and 11B30 (secondary) c The Author(s) 2017. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.93, on 28 Sep 2021 at 20:53:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/fms.2017.12 E. Naslund and W. Sawin 2 1. Introduction A collection of k sets is said to form a k-sunflower, or ∆-system, if the intersection of any two sets from the collection is the same. A family of sets F is said to be k-sunflower-free if no k members form a k-sunflower, and when k D 3 we simply say that the collection F is sunflower-free. It is a longstanding conjecture that sunflower-free families must be small, and there are two natural situations in which we may ask this question. The first, and most general case, is when each set in the family has size m. Erdos˝ and Rado made the following conjecture which is now known as the Sunflower Conjecture. CONJECTURE 1 (Erdos–Rado˝ Sunflower Conjecture [5]). Let k > 3, and suppose that F is a k-sunflower-free family of sets, each of size m. Then j j m F 6 Ck for a constant Ck > 0 depending only on k. In their paper, Erdos˝ and Rado [5] proved that any k-sunflower-free family of m m sets of size m has size at most mW.k − 1/ , but the conjectured bound of Ck remains out of reach for any k > 3. The second setting for upper bounds for k-sunflower-free sets concerns the case where each member of F is a subset of the same n-element set. There can be at most 2n such subsets, and the Erdos–Szemer˝ edi´ sunflower conjecture states that this trivial upper bound can be improved by an exponential factor. CONJECTURE 2 (Erdos–Szemer˝ edi´ sunflower conjecture [6]). Let S be a k- sunflower-free collection of subsets of f1; 2;:::; ng. Then n jSj < ck for some constant ck < 2 depending only on k. In Erdos˝ and Szemeredi’s´ paper [6], they prove that Conjecture2 follows from Conjecture1 (see also [ 1, Theorem 2.3]), and so it is a weaker variant of the sunflower problem. Let Fk .n/ denote the size of the largest k-sunflower-free collection F of subsets of f1; 2;:::; ng, and define S 1=n µk D lim sup Fk .n/ n!1 S to be the Erdos–Szmer˝ edi-k-sunflower-free´ capacity. The trivial bound is µk 6 2, S and the Erdos–Szemer˝ edi´ sunflower conjecture states that µk < 2 for all k > 3. Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.93, on 28 Sep 2021 at 20:53:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/fms.2017.12 Upper bounds for sunflower-free sets 3 S In this paper we prove new bounds for the sunflower-free capacity µ3 . It is a theorem of Alon et al. [1, page 7] that the recent work of Ellenberg and n S Gijswijt [4] on progression-free sets in F3 implies that µ3 < 2, and before this, the bestp upper bound for a sunflower-free collection of f1; 2;:::; ng was n − 2 exp c n due to Erdos˝ and Szemeredi´ [6]. We givep a simple proof of a S C quantitative version of [1, page 7], showing that µ3 6 1 C where C is the capset capacity. However, using the ideas from the recent breakthrough of n Ellenberg and Gijswijt and Croot et al. [3, 4] on progressions in F3, and from Tao’s version of the argument [8], we apply the polynomial method directly to this problem, and obtain a stronger result: THEOREM 1. Let F be a sunflower-free collection of subsets of f1; 2;:::; ng. Then X n jFj 3.n C 1/ ; 6 k k6n=3 and 3 µS D 1:889881574 ··· 3 6 22=3 There is still a large gap between upper and lower bounds for the sunflower-free S capacity µ3 . In Section3 we turn to the sunflower problem in the set f1; 2;:::; Dg, which we always think of as Z=DZ . Alon, Shpilka and Umans [1, Definition 2.5], defined n a k-sunflower in .Z=DZ/ for k 6 D to be a collection of k vectors such that in each coordinate they are either all different or all the same. When k D 3 and D D 3 this condition is equivalent to being a three-term arithmetic progression n in F3. CONJECTURE 3 (Sunflower conjecture in .Z=DZ/). Let k 6 D, and let A ⊂ .Z=DZ/n be a k-sunflower-free set. Then j j n A 6 bk for a constant bk depending only on k. The motivation for this problem comes from [1, Theorem 2.6] where they proved that Conjecture3 is equivalent to the Erd os–Rado˝ sunflower conjecture. In particular, if there exists a constant b3 independent of D such that any 3-sunflower- n n free set in .Z=DZ/ has size at most b3 , then Conjecture1 holds for k D 3 with c3 D e · b3. Since a sunflower-free set cannot contain a three-term arithmetic progression, the recent result of Ellenberg and Gijswijt [4] implies an upper bound Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.93, on 28 Sep 2021 at 20:53:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/fms.2017.12 E. Naslund and W. Sawin 4 ⊂ n j j n for sunflower-free sets A .Z=DZ/ for D prime of the form A 6 cD, where −I ..D−1/=3/ cD D De for a function I defined in [4] in terms of a certain optimization problem. It is not too hard to see that c 0 < lim D < 1: D!1 D Using the characters χ V Z=DZ ! C instead of polynomials, we prove the following theorem: n THEOREM 2. Let D > 3, and let A ⊂ .Z=DZ/ be a sunflower-free set. Then j j n A 6 cD D 3 − 2=3 where cD 22=3 .D 1/ : This can be seen as progress towards the Erdos–Rado˝ sunflower conjecture, and we remark that the now resolved Erdos–Szemer˝ edi´ conjecture for k D 3 is 1− equivalent to proving that cD < D for some and all D sufficiently large [1, Theorem 2.7]. To prove Theorems1 and2 we bound the slice rank of a function of three variables T .x; y; z/ which is nonvanishing if and only if x D y D z or x; y; z form a sunflower. A function f V Ak ! F, where Ak D A × A × · · · × A denotes the Cartesian product and F is a field, is said to be a slice if it can be written in the form f .x1;:::; xk / D h.xi /g.x1;:::; xi−1; xiC1;:::; xk / − where h V A ! F and g V Ak 1 ! F.
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