Reducts of random hypergraphs

Abstract

For each k ⩾ 1, let Γ k be the countable universal homogeneous k-hypergraph. In this paper, we shall classify the closed permutation groups G such that Aut( Γ k ) ⩽ G ⩽ Sym( Γ k ). In particular, we shall show that there exist only finitely many such groups G for each k ⩾ 1. We shall also show that each of the associated reducts of Γ k is homogeneous with respect to a finite relational language.