Distinguishing partitions and asymmetric uniform hypergraphs

Abstract

A distinguishing partition for an action of a group Γ on a set X is a partition of X that is preserved by no nontrivial element of Γ. As a special case, a distinguishing partition of a graph is a partition of the vertex set that is preserved by no nontrivial automorphism. In this paper we provide a link between distinguishing partitions of complete equipartite graphs and asymmetric uniform hypergraphs. Suppose that m ≥ 1 and n ≥ 2. We show that an asymmetric n-uniform hypergraph with m edges exists if and only if m ≥ f(n), where f(2) = f(14) = 6, f(6) = 5, and f(n)= ⌊ log₂(n + 1) ⌋ + 2 otherwise. It follows that a distinguishing partition of K_m(n) = K_{n, n, ..., n}, or equivalently for the wreath product action S_n Wr S_m, exists if and only if m ≥ f(n).