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 k-uniform hypergraph of order n, where k = 2` or k = 2` + 1 and n ≡ 1 (mod 2`+1), then the highest power of any prime dividing n must be congruent to 1 modulo 2`+1. We show that this necessary condition is also sufficient in many cases – for example, for n a prime power, and for k = 3 and n odd – thus generalizing… CONTINUE READING