On cross t-intersecting families of sets

Abstract. Two hypergraphs H 1 , H 2 are called cross-intersecting if e 1 ∩ e 2 = ∅ for every pair of edges e 1 ∈ H 1 , e 2 ∈ H 2 . Each of the hypergraphs is then said to block the other. Given parameters n, r, m we determine the maximal size of a sub-hypergraph of [n] r (meaning that it is r-partite, with all sides of size n) for which there exists a blocking sub-hypergraph of [n] r of size m. The answer involves a fractal-like (that is, self-similar) sequence, first studied by Knuth. We also study the same question with n r replacing [n] r .1. Blockers in r-partite hypergraphs 1.1. Blockers. For a set A and a number r let A r be the set of all subsets of size r of A. Given numbers r and n let [n] = {1, 2, . . . , n}, and let [n] r be the complete r-partite hypergraph with all sides being equal to [n]. Let U be either [n] r or [n] r , and let F be a sub-hypergraph of U . The blocker B(F ) of F is the set of those edges of U that meet all edges of F . For a number t we denote by b(t) the maximal size of |B(F )|, where F ranges over all sets of t edges in U . Which of the two meanings of "b(t)" we are using will be clear from the context. Background -the Erdős-Ko-Rado theorem and rainbow matchings.A matching is a collection of disjoint sets. The largest size of a matching in a hypergraph H is denoted by ν(H). The famous Erdős-Ko-Rado (EKR) theorem [8] states that if r ≤ n 2 and a hypergraph H ⊆ [n] r has more than n−1 r−1 edges, then ν(H) > 1. This has been extended in more than one way to pairs of hypergraphs. For example, in [17,19] the following was proved: Theorem 1.1. If r ≤ n 2 , and2 (in particular if |H i | > n−1 r−1 , i = 1, 2), then there exist disjoint edges, e 1 ∈ H 1 , e 2 ∈ H 2 .In [17] this was also extended to hypergraphs of different uniformities. In [20,21] a version of this result was proved for t-intersecting pairs of hypergraphs, for large enough n.

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“…In [20,21] a version of this result was proved for t-intersecting pairs of hypergraphs, for large enough n.…”

Section: Background -The Erdős-ko-rado Theorem and Rainbow Matchingsmentioning

confidence: 96%

Cross-intersecting pairs of hypergraphs

Aharoni

Howard

2017

Journal of Combinatorial Theory, Series A

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“…If |F| = m and its u-cascade form is given by (11), then it follows from the Kruskal-Katona Theorem [4,3] that…”

Section: Preliminariesmentioning

confidence: 99%

“…A simple computation shows that m(α, β) = 1 if α + β > 1, and m(α, β) = 1 4 if α + β = 1. In [11] it is shown that if max{α, β} ≤ 1 2 then m(n, α, β) = αβ…”

Section: Introductionmentioning

confidence: 99%

When are stars the largest cross-intersecting families?

Tokushige

2020

Discrete Mathematics

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“…, A k of a given family F. The paper [14] analyses this problem in general, particularly showing that, for k sufficiently large, both the sum and the product are maxima if A 1 = · · · = A k = L for some largest t-intersecting subfamily L of F. Therefore, this problem incorporates the t-intersection problem. Solutions have been obtained for various families (see [14]), including [n] r (see [5,7,25,29,39,43,46,48,49]), 2 [n] (see [14,38]), [n] r (see [15]), families of integer sequences [6,16,25,41,42,47,49,51], and families of vector spaces [44]. Most of these results tell us that, for the family F under consideration and for certain values of k, the sum or the product is maximum when A 1 = · · · = A k = L for some largest t-star L of F. In such a case, L is a largest t-intersecting subfamily of F. Remark 1.…”

Section: Introductionmentioning

confidence: 99%

The maximum product of weights of cross-intersecting families

Borg

2016

J. London Math. Soc.

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Two families A and B of sets are said to be cross-t-intersecting if each set in A intersects each set in B in at least t elements. An active problem in extremal set theory is to determine the maximum product of sizes of cross-t-intersecting subfamilies of a given family. We prove a cross-t-intersection theorem for weighted subsets of a set by means of a new subfamily alteration method, and use the result to provide solutions for three natural families. For r ∈ [n] = {1, 2, . . . , n}, letbe the family of r-element subsets of [n], and let [n] ≤r be the family of subsets of [n] that have at most r elements. Let F n,r,t be the family of sets in [n] ≤r that contain [t]. We show that if g : g(C)D∈Fn,s,t h(D), and equality holds if A = F m,r,t and B = F n,s,t . We prove this in a more general setting and characterise the cases of equality. We use the result to show that the maximum product of sizes of two cross-t-intersecting families A ⊆ [m] r and B ⊆ [n] s is m−t r−t n−t s−t for min{m, n} ≥ n 0 (r, s, t), where n 0 (r, s, t) is close to best possible. We obtain analogous results for families of integer sequences and for families of multisets. The results yield generalisations for k ≥ 2 cross-t-intersecting families, and Erdos-Ko-Rado-type results.

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On cross t-intersecting families of sets

Abstract: For all p, t with 0 < p < 0.11 and 1 t 1/(2p), there exists n 0 such that for all n, k with n > n 0 and k/n = p the following holds: if A and B are k-uniform families on n vertices, and |A ∩ B| t holds for all A ∈ A and B ∈ B, then |A ||B| n−t k−t 2 .

Cited by 19 publications

References 9 publications

Cross-intersecting pairs of hypergraphs

Cross-intersecting pairs of hypergraphs

When are stars the largest cross-intersecting families?

The maximum product of weights of cross-intersecting families

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