Reducts of random hypergraphs

Abstract

For each k ⩾ 1, let Γk be the countable universal homogeneous k-hypergraph. In this paper, we shall classify the closed permutation groups G such that Aut(Γk) ⩽ G ⩽ Sym(Γk). 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 Γk is homogeneous with respect to a finite relational language.