5 research outputs found
Partition-crossing hypergraphs
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
For a finite set , we say that a set crosses a partition
of if intersects partition
classes. If , this means that meets all classes , whilst
for the elements of the crossing set belong to mutually
distinct classes. A set system crosses , if so does some
. The minimum number of -element subsets, such that every
-partition of an -element set is crossed by at least one of them, is
denoted by .
The problem of determining these minimum values for 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
for every fixed as [Graphs Combin., 25 (2009),
807--816]. Here we consider the more general problem for two parameters and
, and establish lower and upper bounds for . For various
combinations of the three values we obtain asymptotically tight
estimates, and also point out close connections of the function to
Tur\'an-type extremal problems on graphs and hypergraphs, or to balanced
incomplete block designs.Comment: 15 page