Ultrahomogeneous hypertournaments

Abstract

A $k$-hypertournament is a higher-arity generalisation of a tournament, consisting of a set together with an irreflexive $k$-ary relation such that the automorphism group of each substructure of size $k$ is the alternating group $\mathrm{A}_k$. We show that, for each $k \geq 3$, there are continuum-many non-isomorphic countable ultrahomogeneous $k$-hypertournaments, answering a question of Cherlin.