Counting Intersecting and Pairs of Cross-Intersecting Families
نویسندگان
چکیده
منابع مشابه
Counting Intersecting and Pairs of Cross-Intersecting Families
A family of subsets of {1, . . . , n} is called intersecting if any two of its sets intersect. A classical result in extremal combinatorics due to Erdős, Ko, and Rado determines the maximum size of an intersecting family of k-subsets of {1, . . . , n}. In this paper we study the following problem: how many intersecting families of k-subsets of {1, . . . , n} are there? Improving a result of Bal...
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Let A and B denote two families of subsets of an n-element set. The pair (A,B) is said to be `-cross-intersecting iff |A∩B| = ` for all A ∈ A and B ∈ B. Denote by P`(n) the maximum value of |A||B| over all such pairs. The best known upper bound on P`(n) is Θ(2), by Frankl and Rödl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n ≥ 2`, a simple construction of an `-cross-intersectin...
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Frankl, P., On cross-intersecting families, Discrete Mathematics 108 (1992) 291-295. Let n 3 t z 1 be integers. Let 9, YI be families of subsets of the n-element set X. They are called cross t-intersecting if IF n GI 2 t holds for all F E 9 and G E 3. If 9 = CfI then 9 is called t-intersecting. Let m(n, t) denote the maximum possible cardinality of a r-intersecting family. Our main result says ...
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Two hypergraphs H1, H2 are called cross-intersecting if e1 ∩ e2 ̸= ∅ for every pair of edges e1 ∈ H1, e2 ∈ H2. Each of the hypergraphs is then said to block the other. Given integers 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 sel...
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Let F be a family of pairs of sets. We call it an (a, b)-set system if for every set-pair (A,B) in F we have that |A| = a, |B| = b, A ∩ B = ∅. The following classical result on families of cross-intersecting set-pairs is due to Bollobás [6]. Let F be an (a, b)-set system with the cross-intersecting property, i.e., for (Ai, Bi), (Aj, Bj) ∈ F with i 6= j we have that both Ai ∩ Bj and Aj ∩ Bi are ...
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ژورنال
عنوان ژورنال: Combinatorics, Probability and Computing
سال: 2017
ISSN: 0963-5483,1469-2163
DOI: 10.1017/s0963548317000529