5 research outputs found

    Partition-crossing hypergraphs

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    For a finite set X, we say that a set H ⊆ X crosses a partition P = (X1, . . . , Xk) of X if H intersects min(|H|, k) partition classes. If |H| ≥ k, this means that H meets all classes Xi, whilst for |H| ≤ k the elements of the crossing set H belong to mutually distinct classes. A set system H crosses P, if so does some H ∈ H. The minimum number of r-element subsets, such that every k-partition of an n-element set X is crossed by at least one of them, is denoted by f(n, k, r). The problem of determining these minimum values for k = r was raised and studied by several authors, first by Sterboul in 1973 [Proc. Colloq. Math. Soc. J. Bolyai, Vol. 10, Keszthely 1973, North-Holland/American Elsevier, 1975, pp. 1387–1404]. The present authors determined asymptotically tight estimates on f(n, k, k) for every fixed k as n → ∞ [Graphs Combin., 25 (2009), 807–816]. Here we consider the more general problem for two parameters k and r, and establish lower and upper bounds for f(n, k, r). For various combinations of the three values n, k, r we obtain asymptotically tight estimates, and also point out close connections of the function f(n, k, r) to Tur´an-type extremal problems on graphs and hypergraphs, or to balanced incomplete block designs

    Partition-Crossing Hypergraphs

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    For a finite set XX, we say that a set HXH\subseteq X crosses a partition P=(X1,,Xk){\cal P}=(X_1,\dots,X_k) of XX if HH intersects min(H,k)\min (|H|,k) partition classes. If Hk|H|\geq k, this means that HH meets all classes XiX_i, whilst for Hk|H|\leq k the elements of the crossing set HH belong to mutually distinct classes. A set system H{\cal H} crosses P{\cal P}, if so does some HHH\in {\cal H}. The minimum number of rr-element subsets, such that every kk-partition of an nn-element set XX is crossed by at least one of them, is denoted by f(n,k,r)f(n,k,r). The problem of determining these minimum values for k=rk=r was raised and studied by several authors, first by Sterboul in 1973 [Proc. Colloq. Math. Soc. J. Bolyai, Vol. 10, Keszthely 1973, North-Holland/American Elsevier, 1975, pp. 1387--1404]. The present authors determined asymptotically tight estimates on f(n,k,k)f(n,k,k) for every fixed kk as nn\to \infty [Graphs Combin., 25 (2009), 807--816]. Here we consider the more general problem for two parameters kk and rr, and establish lower and upper bounds for f(n,k,r)f(n,k,r). For various combinations of the three values n,k,rn,k,r we obtain asymptotically tight estimates, and also point out close connections of the function f(n,k,r)f(n,k,r) to Tur\'an-type extremal problems on graphs and hypergraphs, or to balanced incomplete block designs.Comment: 15 page

    Acta Cybernetica : Volume 23. Number 3.

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    Partition-Crossing Hypergraphs

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