Almost Cross-Intersecting and Almost Cross-Sperner Pairs of Families of Sets

Gerbner, Dániel; Lemons, Nathan; Palmer, Cory; Pálvölgyi, Dömötör; Patkós, Balázs; Szécsi, Vajk

doi:10.1007/s00373-012-1138-2

Cited by 4 publications

(8 citation statements)

References 10 publications

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“…If H is a complete multipartite subgraph of the complement of Kneser(n, k) such that no colour class contains more than p|H| vertices, then |H| ≤ n−1 k−1 . Note that similar, but incomparable, generalisations of the Erdős-Ko-Rado Theorem have recently been explored in [5,4,18]. Theorem 3 is proven in Section 4, since it follows almost directly from our proof of the lower bound on the treewidth of a Kneser graph.…”

mentioning

confidence: 84%

Treewidth of the Kneser Graph and the Erdős-Ko-Rado Theorem

Harvey

Wood

2014

Electron. J. Combin.

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Treewidth is an important and well-known graph parameter that measures the complexity of a graph. The Kneser graph Kneser(n, k) is the graph with vertex set [n] k , such that two vertices are adjacent if they are disjoint. We determine, for large values of n with respect to k, the exact treewidth of the Kneser graph. In the process of doing so, we also prove a strengthening of the Erdős-Ko-Rado Theorem (for large n with respect to k) when a number of disjoint pairs of k-sets are allowed.

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confidence: 84%

Treewidth of the Kneser Graph and the Erdős-Ko-Rado Theorem

Harvey

Wood

2014

Electron. J. Combin.

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“…Finally, in Section 4, we consider the range of large p. Using the result of [11] stated above, we know that if C p,q is a submatrix of A k,t of size n × n, then n ≤ (2q − 1) 2t t , and [23] proved a conjecture of [12] and showed that for large enough q and t, the size of a q-almost intersecting family F is bounded by (q + 1) 2t−2 t−1 . Note that this last result refers to q-almost cross intersecting pairs (F, G) in which F = G. Furthermore, the constructions presented in [23], which achieve this bound, do not have a circulant intersection matrix.…”

Section: Our Resultsmentioning

confidence: 99%

“…Another variation of the problem of cross intersecting families was introduced by Gerbner et al [11], which defined the notion of a q-almost cross intersecting pair (F, G). Here every subset in F does not intersect with exactly q subsets in G and vice versa.…”

Section: Introductionmentioning

confidence: 99%

“…Using a classic theorem of Bollobás [3], it is possible to prove that the largest square submatrix C of A k,t , representing a 1-almost cross intersecting pair, is of size 2t t × 2t t . A theorem proved in [11] shows that if C is a submatrix of size n × n of A k,t , with exactly q zeros in each row and column, then n ≤ (2q − 1) 2t t . In this paper we consider the problem of finding maximal circulant submatrices of A k,t , representing an almost cross intersecting pair, for a range of parameters.…”

Section: Introductionmentioning

confidence: 99%

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Circulant almost cross intersecting families

Parnas

2022

Art Discrete Appl. Math.

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Let F and G be two t-uniform families of subsets over [k] = {1, 2, ..., k}, where |F| = |G|, and let C be the adjacency matrix of the bipartite graph whose vertices are the subsets in F and G, where there is an edge between A ∈ F and B ∈ G if and only if A ∩ B = ∅. The pair (F, G) is q-almost cross intersecting if every row and column of C has exactly q zeros.We further restrict our attention to q-almost cross intersecting pairs that have a circulant intersection matrix C p,q , determined by a column vector with p > 0 ones followed by q > 0 zeros. This family of matrices includes the identity matrix in one extreme, and the adjacency matrix of the bipartite crown graph in the other extreme.We give constructions of pairs (F, G) whose intersection matrix is C p,q , for a wide range of values of the parameters p and q, and in some cases also prove matching upper bounds. Specifically, we prove results for the following values of the parameters:(2) 2t ≤ p ≤ t 2 and any q > 0, where k ≥ p + q.(3) p that is exponential in t, for large enough k.Using the first result we show that if k ≥ 4t − 3 then C 2t−1,k−2t+1 is a maximal isolation submatrix of size k × k in the 0, 1-matrix A k,t , whose rows and columns are labeled by all subsets of size t of [k], and there is a one in the entry on row x and column y if and only if subsets x, y intersect.

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“…By contrast, it was also shown in [11] that the maximum size of a k-uniform s-almost intersecting hypergraph does not grow with the size of the ground set: every k-uniform s-almost intersecting hypergraph has at most s 2ks ks edges. Gerbner, Lemons, Palmer, Pálvölgyi, Patkós and Szécsi [10] subsequently improved this bound to (2s − 1) 2k…”

Section: Introductionmentioning

confidence: 97%

Hypergraphs of Bounded Disjointness

Scott¹,

Wilmer²

2014

SIAM J. Discrete Math.

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A k-uniform hypergraph is s-almost intersecting if every edge is disjoint from exactly s other edges. Gerbner, Lemons, Palmer, Patkós and Szécsi conjectured that for every k, and s > s 0 (k), every k-uniform s-almost intersecting hypergraph has at most (s + 1) 2k−2 k−1 edges. We prove a strengthened version of this conjecture and determine the extremal graphs. We also give some related results and conjectures.

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Almost Cross-Intersecting and Almost Cross-Sperner Pairs of Families of Sets

Cited by 4 publications

References 10 publications

Treewidth of the Kneser Graph and the Erdős-Ko-Rado Theorem

Treewidth of the Kneser Graph and the Erdős-Ko-Rado Theorem

Circulant almost cross intersecting families

Hypergraphs of Bounded Disjointness

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