Vertex-transitive self-complementary uniform hypergraphs

In this paper we examine the orders of vertex-transitive self-complementary uniform hypergraphs. In particular, we prove that if there exists a vertex-transitive self-complementary kk-uniform hypergraph of order nn, where k=2ℓk=2ℓ or k=2ℓ+1k=2ℓ+1 and n≡1(mod2ℓ+1), then the highest power of any prime dividing nn must be congruent to 1 modulo 2ℓ+12ℓ+1. We show that this necessary condition is also sufficient in many cases–for example, for nn a prime power, and for k=3k=3 and nn odd–thus generalizing the result on vertex-transitive self-complementary graphs of Rao and Muzychuk. We also give sufficient conditions for the existence of vertex-transitive self-complementary uniform hypergraphs in several other cases. Since vertex-transitive self-complementary uniform hypergraphs are equivalent to a certain kind of large sets of tt-designs, the results of the paper imply the corresponding results in design theory.