Multiply intersecting families of sets

Authors:
Zoltán Füredi;Zsolt Katona
Affiliations:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL and Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, Budapest 1364, Hungary;Department of Computer Science, Eötvös Loránd University, Mexikói út 31/a, Budapest 1145, Hungary
Venue:
Journal of Combinatorial Theory Series A
Year:
2004

Citing 8
Cited 1

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Multiply-intersecting families

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Proof of a conjecture of Frankl and Füredi

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Intersecting families of sets, no l containing two common elements

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Note: on k-wise set-intersections and k-wise Hamming-distances

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Abstract

Let [n] denote the set {1, 2, ...,n}, 2[n] the collection of all subsets of [n] and F ⊂ 2[n] be a family. The maximum of |F| is studied if any r subsets have an at least s-element intersection and there are no ρ subsets containing t + 1 common elements. We show that |F| ≤ Σi = 0t - s (n - s i) + t + ρ - s/t + 2 - s(n - s t + 1 - s) + ρ - 2 and this bound is asymptotically the best possible as n → ∞ and t ≥ 2s ≥ 2, r, ρ ≥ 2 are fixed.