Self-Complementary Non-Uniform Hypergraphs

Let V be a finite set. For a nonempty subset K of positive integers, a K-hypergraph on V is a hypergraph with vertex set V and edge set E = boolean OR(k is an element of K) E-k , where E-k is a nonempty set of k-subsets of V. We define the complement of a K-hypergraph (V, E) to be the K-hypergraph on V whose edge set consists of the subsets of V with cardinality in K which do not lie in E. A K-hypergraph is called self-complementary if it is isomorphic to its complement. The two extreme classes of self-complementary K-hypergraphs have been studied previously. When |K| = 1 these are the self-complementary uniform hypergraphs, and when |K| = |V| - 1, these are the so called 'self-complementary hypergraphs' studied by Zwonek. In this paper we determine necessary conditions on the order of self-complementary K-hypergraphs, and on the order of regular or vertex-transitive self-complementary K-hypergraphs, for various sets of positive integers K. We also present several constructions for K-hypergraphs to show that these necessary conditions are sufficient for certain sets K. In the language of design theory, the t-subset-regular self-complementary K-hypergraphs correspond to large sets of two isomorphic t-wise balanced designs, or t-partitions, in which the block sizes lie in the set K. Hence the results of this paper imply results in design theory.