Intersecting families of discrete structures are typically trivial
نویسندگان
چکیده
منابع مشابه
Intersecting families of discrete structures are typically trivial
The study of intersecting structures is central to extremal combinatorics. A family of permutations F ⊂ Sn is t-intersecting if any two permutations in F agree on some t indices, and is trivial if all permutations in F agree on the same t indices. A k-uniform hypergraph is tintersecting if any two of its edges have t vertices in common, and trivial if all its edges share the same t vertices. Th...
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Let n,r and t be positive integers. A family F of subsets of [n]={1,2, . . . ,n} is called r-wise t-intersecting if |F1∩·· ·∩Fr|≥ t holds for all F1, . . . ,Fr ∈F . An r-wise 1-intersecting family is also called an r-wise intersecting family for short. An r-wise t-intersecting family F is called non-trivial if |⋂F∈F F |<t. Let us define the Brace–Daykin structure as follows. F BD = {F ⊂ [n] : |...
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For a family F of sets, let μ(F) denote the size of a smallest set in F that is not a subset of any other set in F , and for any positive integer r, let F (r) denote the family of r-element sets in F . We say that a family A is of Hilton-Milner (HM ) type if for some A ∈ A, all sets in A\{A} have a common element x / ∈ A and intersect A. We show that if a hereditary family H is compressed and μ...
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It is well known that an intersecting family of subsets of an nelement set can contain at most 2n−1 sets. It is natural to wonder how ‘close’ to intersecting a family of size greater than 2n−1 can be. Katona, Katona and Katona introduced the idea of a ‘most probably intersecting family.’ Suppose that A is a family and that 0 < p < 1. Let A(p) be the (random) family formed by selecting each set ...
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A family J of subsets of {1, . . . , n} is called a j-junta if there exists J ⊆ {1, . . . , n}, with |J | = j, such that the membership of a set S in J depends only on S ∩ J . In this paper we provide a simple description of intersecting families of sets. Let n and k be positive integers with k < n/2, and let A be a family of pairwise intersecting subsets of {1, . . . , n}, all of size k. We sh...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 2015
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2015.01.003