Partition-crossing hypergraphs

For a finite set $X$, we say that a set $H\subseteq X$ crosses a partition ${\cal P}=(X_1,\dots,X_k)$ 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 ${\cal H}$ crosses ${\cal P}$, if so does some $H\in {\cal H}$. 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 Tur\'an-type extremal problems on graphs and hypergraphs, or to balanced incomplete block designs.