Cross-intersecting pairs of hypergraphs

Two hypergraphs H1,H2 are called cross-intersecting if e1∩e2≠∅ for every pair of edges e1∈H1,e2∈H2. Each of the hypergraphs is then said to block the other. Given integers 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 self-similar sequence, first studied by Knuth. We also study the same question with (nr) replacing [n]r. These results yield new proofs of some known Erdős-Ko-Rado type theorems.