Transitive Extensions of Automorphism Groups of Generic Structures

Abstract

This work addresses the existence of transitive extensions of certain infinite permutation groups which arise as the automorphism groups of model-theoretic structures which are generic in the Fra\"iss\'e sense. The study of transitive extensions has hitherto largely concerned itself with finite permutation groups. Moving beyond the finite realm, we develop combinatorial tools to prove that transitive extensions exist for edge-colored k-hypergraphs only when the number of colors is a power of two and that transitive extensions exist for k-hypertournaments (in the Cherlin sense) only when k is even, among other results.