Cross-intersecting pairs of hypergraphs

Abstract

Two hypergraphs $H_1,\ H_2$ are called {\em cross-intersecting} if $e_1 \cap e_2 \neq \emptyset$ for every pair of edges $e_1 \in H_1,~e_2 \in H_2$. Each of the hypergraphs is then said to {\em block} the other. Given parameters $n,r,m$ we determine the maximal size of a sub-hypergraph of $[n]^r$ (meaning that it is $r$-partite, with all sides of size $n$) for which there exists a blocking sub-hypergraph of $[n]^r$ of size $m$. The answer involves a fractal-like (that is, self-similar) sequence, first studied by Knuth. We also study the same question with $\binom{n}{r}$ replacing $[n]^r$.