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 (n r ) replacing [n]r. These results yield new proofs of some known Erdős-Ko-Rado type theorems. 1. Blockers in r-partite hypergraphs 1.1. Blockers. For a set A and a number r let ( A r ) be the set of all subsets of size r of A, in other words the complete r-uniform hypergraph on A. Given numbers r and n let [n] = {1, 2, . . . , n}, and let [n] be the complete r-partite hypergraph with all sides being equal to [n]. Let U be either ( [n] r ) or [n], and let F be a sub-hypergraph of U . The blocker B(F ) = B(U,F ) of F is the set of those edges of U that meet all edges of F . For a number t we denote by bp(t) (resp. bc(t) reference to the uniformity r is suppressed in both notations) the maximal size of |B([n], F )| (resp. |B( ( [n] r ) , F )| ), where F ranges over all sets of t edges in [n] (resp. ( [n] r ) ). The subscript p alludes at “partite”, and the subscript c alludes at “complete”. The aim of this paper is to calculate bp(t) and bc(t) for all values of n, r and t. As a side benefit, this will enable us to give new proofs of some well-known Erdős-Ko-Rado type results. 1.2. Cross intersecting versions of the Erdős-Ko-Rado theorem. The famous Erdős-Ko-Rado (EKR) theorem [9] states that if r ≤ n2 and a hypergraph H ⊆ ( [n] r ) has more than ( n−1 r−1 ) edges, then H contains two disjoint sets. Many extensions of this theorem have been proved for pairs of hypergraphs. In [17, 20] the following was proved: Theorem 1.1. If r ≤ n2 , and H1,H2 ⊆ ( [n] r ) satisfy |H1||H2| > ( n−1 r−1 )2 (in particular if |Hi| > ( n−1 r−1 ) , i = 1, 2), then there exist disjoint edges, e1 ∈ H1, e2 ∈ H2. In [17] this was also extended to hypergraphs of different uniformities. Versions of this result were proved for cross t-intersecting pairs of hypergraphs, in [13, 21, 23]. The EKR theory has been also extended to sets living in [n], rather than ( [n] r ) . An easy observation is that any subset of [n] of size larger than nr−1 contains two disjoint edges. This can be proved from the fact that [n] is the union of nr−1 perfect matchings. More interesting are cross-intersecting type results: Theorem 1.2. A pair F1, F2 of subsets of [n] r satisfying |F1| > nr−1 and |F2| ≥ nr−1 has a rainbow matching. And the even stronger: Theorem 1.3. If F1, F2 ⊆ [n] and |F1||F2| > n2(r−1) then the pair (F1, F2) has a rainbow matching. Theorem 1.3 was proved in [18]. It was generalized to cross t-intersecting pairs of hypergraphs and to hypregaphs of different uniformities in [1, 3, 4, 13, 19, 22] ([1, 22] use spectral methods). At the end of the next section we shall use the techniques of the present paper to give new proofs for these results. The research of the first author was supported by BSF grant no. 2006099, by GIF grant no. I −879 − 124.6/2005, by the Technion’s research promotion fund, and by the Discont Bank chair. The research of the second author was supported by BSF grant no. 2006099, and by ISF grants Nos. 779/08, 859/08 and 938/06.