Bipartite Hansel results for hypergraphs

Abstract

For integers n≥k≥2, let V be an n-element set, and let Vk denote the set of all k-element subsets of V. For disjoint A,B⊆V, we say {A,B} covers K∈Vk if K⊆A∪̇B and K meets each of A and B, i.e., K∩A≠0̸≠K∩B. We say that a collection C of such pairs {A,B} covers Vk if every element of Vk is covered by at least one member of C. When k=2, such a family is called a separating system of V, where this concept was introduced by Rényi (1961) and studied by many authors.Let h(n,k) denote the minimum value of ∑{A,B}∈C(|A|+|B|) among all covers C of Vk. Hansel (1964) determined the bounds ⌈nlog2n⌉≤h(n,2)≤n⌈log2n⌉, and Bollobás and Scott (2007) determined an exact formula for h(n,2). We extend these results to give an exact formula for h(n,k), and to guarantee that all optimal covers C of Vk share a common degree-sequence. Our proofs follow lines of Bollobás and Scott, together with weight-shifting arguments in a similar vein to some of Motzkin and Straus (1965).