Erdös-Ko-Rado from Kruskal-Katona

Daykin, D. E.

doi:10.1016/0097-3165(74)90013-2

Cited by 61 publications

(50 citation statements)

References 5 publications

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“…The proof involves combining the methods above with an idea of Daykin [6]. First we remark that if r l m and G is an r-graph with K r l (G) = x l then K r m (G)…”

Section: Stability For Intersecting Families Imentioning

confidence: 99%

Shadows and intersections: Stability and new proofs

Keevash

2008

Advances in Mathematics

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We give a short new proof of a version of the Kruskal-Katona theorem due to Lovász. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi by means of a purely combinatorial result of Frankl. We also give an algebraic perspective on these problems, giving yet another proof of intersection stability that relies on expansion of a certain Cayley graph of the symmetric group, and an algebraic generalisation of Lovász's theorem that answers a question of Frankl and Tokushige.

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“…The proof involves combining the methods above with an idea of Daykin [6]. First we remark that if r l m and G is an r-graph with K r l (G) = x l then K r m (G)…”

Section: Stability For Intersecting Families Imentioning

confidence: 99%

Shadows and intersections: Stability and new proofs

Keevash

2008

Advances in Mathematics

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“…For another slight generalization, we denote by I(A) the family of isolated points in A, that is, I(A) = {a ∈ A : (a, b) / ∈ E(Kn(n, k)) for all b ∈ A}. In his paper, Borg [1] extended Daykin's proof [2] of the EKR theorem to obtain the following improvement:…”

Section: Similar Resultsmentioning

confidence: 99%

Erdös-Ko-Rado from intersecting shadows

Katona

Kisvölcsey

2012

Discuss. Math. Graph Theory

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A set system is called t-intersecting if every two members meet each other in at least t elements. Katona determined the minimum ratio of the shadow and the size of such families and showed that the Erdős-Ko-Rado theorem immediately follows from this result. The aim of this note is to reproduce the proof to obtain a slight improvement in the Kneser graph. We also give a brief overview of corresponding results.

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“…Daykin [4] showed how the EKR theorem can be derived from the Kruskal-Katona theorem. His proof also yields the case r = 2 of Conjecture 1.5.…”

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

confidence: 99%

“…The proof uses an already mentioned idea of Daykin [4], who gave a proof of the EKR theorem using the Kruskal-Katona theorem.…”

Section: The Sequence M R (I) Write M = 2rmentioning

confidence: 99%

Cross-intersecting pairs of hypergraphs

Aharoni

Howard

2017

Journal of Combinatorial Theory, Series A

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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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Erdös-Ko-Rado from Kruskal-Katona

Cited by 61 publications

References 5 publications

Shadows and intersections: Stability and new proofs

Shadows and intersections: Stability and new proofs

Erdös-Ko-Rado from intersecting shadows

Cross-intersecting pairs of hypergraphs

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