Forbidden vector‐valued intersections

Keevash, Peter; Long, Eoin

doi:10.1112/plms.12338

Cited by 5 publications

(3 citation statements)

References 29 publications

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“…Over the years this problem has been reiterated several times [7,30] including in a recent collaborative "polymath" project [37]. The case k = 2 of the problem has received considerable attention [22,25,27,33,34,41], partly due to its huge impact in discrete geometry [26], communication complexity [39] and quantum computing [5]. Another case that has a rich history [13,15,16,[19][20][21]24] is t = 0 (a matching of size k is forbidden); the optimal construction in this case is predicted by the Erdős Matching Conjecture.…”

Section: Sunflowersmentioning

confidence: 99%

Unavoidable Hypergraphs

Bucić¹,

Draganić²,

Sudakov³

et al. 2021

Trends in Mathematics

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The following very natural problem was raised by Chung and Erdős in the early 80's and has since been repeated a number of times. What is the minimum of the Turán number ex(n, H) among all r-graphs H with a fixed number of edges? Their actual focus was on an equivalent and perhaps even more natural question which asks what is the largest size of an r-graph that can not be avoided in any r-graph on n vertices and e edges? In the original paper they resolve this question asymptotically for graphs, for most of the range of e. In a follow-up work Chung and Erdős resolve the 3-uniform case and raise the 4-uniform case as the natural next step. In this paper we make first progress on this problem in over 40 years by asymptotically resolving the 4-uniform case which gives us some indication on how the answer should behave in general.

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Section: Sunflowersmentioning

confidence: 99%

Unavoidable Hypergraphs

Bucić¹,

Draganić²,

Sudakov³

et al. 2021

Trends in Mathematics

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“…Over the years this problem has been reiterated several times [7,27] including in a recent collaborative "polymath" project [33]. The case k = 2 of the problem (a certain fixed intersection size is forbidden) is better known as the restricted intersection problem and has received considerable attention over the years [20,24,26,29,30,37], partly due to its huge impact in discrete geometry [25], communication complexity [36] and quantum computing [5]. Another case that has a rich history [12,13,14,17,18,19,23] is t = 0 (a matching of size k is forbidden); the optimal construction in this case is predicted by the famous Erdős Matching Conjecture.…”

Section: Turán Numbers Of Sunflowersmentioning

confidence: 99%

Turán numbers of sunflowers

Domagoj¹,

Bucić²,

Sudakov³

2021

Preprint

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A collection of distinct sets is called a sunflower if the intersection of any pair of sets equals the common intersection of all the sets. Sunflowers are fundamental objects in extremal set theory with relations and applications to many other areas of mathematics as well as theoretical computer science. A central problem in the area due to Erdős and Rado from 1960 asks for the minimum number of sets of size r needed to guarantee the existence of a sunflower of a given size. Despite a lot of recent attention including a polymath project and some amazing breakthroughs, even the asymptotic answer remains unknown.We study a related problem first posed by Duke and Erdős in 1977 which requires that in addition the intersection size of the desired sunflower be fixed. This question is perhaps even more natural from a graph theoretic perspective since it asks for the Turán number of a hypergraph made by the sunflower consisting of k edges, each of size r and with common intersection of size t. For a fixed size of the sunflower k, the order of magnitude of the answer has been determined by Frankl and Füredi. In the 1980's, with certain applications in mind, Chung, Erdős and Graham and Chung and Erdős considered what happens if one allows k, the size of the desired sunflower, to grow with the size of the ground set. In the three uniform case r = 3 the correct dependence on the size of the sunflower has been determined by Duke and Erdős and independently by Frankl and in the four uniform case by Bucić, Draganić, Sudakov and Tran. We resolve this problem for any uniformity, by determining up to a constant factor the n-vertex Turán number of a sunflower of arbitrary uniformity r, common intersection size t and with the size of the sunflower k allowed to grow with n.

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“…The work of Frankl and Rödl has proven to be very influential, and it has found applications in a number of different areas such as discrete geometry [FR90], communication complexity [S99] and quantum computing [BCW99]. Further progress on the Erdős-Sós problem was made by several authors, including the very recent works of Ellis/Keller/Lifshitz [EKL16], Keller/Lifshitz [KLi21] and Kupavskii/Zaharov [KZ22] (see, also, [KLLM21,KLo20] for closely related developments). Collectively, the papers [EKL16,KLi21] obtain the sharp estimate |A| n−(ℓ+1) k−(ℓ+1) for every family A ⊆ [n] k whose intersections forbid ℓ in the regime 2ℓ k 1 2 − ε n with n n 0 (ℓ, ε) for some (unspecified) threshold function n 0 (ℓ, ε).…”

Section: Introductionmentioning

confidence: 99%

Forbidden sparse intersections

Dodos¹,

Karamanlis²

2023

Preprint

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Let n be a positive integer, let 0 < p p ′ 1 2 , and let ℓ pn be a nonnegative integer. We prove that if F , G ⊆ {0, 1} n are two families whose cross intersections forbid ℓ-that is, they satisfy |A ∩ B| = ℓ for every A ∈ F and every B ∈ G-then, setting t := min{ℓ, pn − ℓ}, we have the subgaussian boundwhere µp and µ p ′ denote the p-biased and p ′ -biased measures on {0, 1} n respectively.

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Forbidden vector‐valued intersections

Cited by 5 publications

References 29 publications

Unavoidable Hypergraphs

Unavoidable Hypergraphs

Turán numbers of sunflowers

Forbidden sparse intersections

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