For a finite set ▫$X$▫, we say that a set ▫$H\subseteq X$▫ crosses a partition ▫$\mathcal P=(X_1,\dots,X_k)$▫ of ▫$X$▫ if ▫$H$▫ intersects ▫$\min(|H|, k)$▫ partition classes. If ▫$|H|\geq k$▫, this ...means that ▫$H$▫ meets all classes ▫$X_i$▫, whilst for ▫$|H|\leq k$▫ the elements of the crossing set ▫$H$▫ belong to mutually distinct classes. A set system ▫$\mathcal H$▫ crosses ▫$\mathcal P$▫, if so does some ▫$H \in\mathcal H$▫. 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 ▫$n\to\infty$▫ [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 Turán-type extremal problems on graphs and hypergraphs, or to balanced incomplete block designs.
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