Transitive Extensions of Automorphism Groups of Generic Structures

TL;DR

This paper investigates conditions for the existence of transitive extensions of automorphism groups of generic structures, revealing combinatorial constraints related to hypergraphs and hypertournaments.

Contribution

It develops new combinatorial tools to analyze transitive extensions of automorphism groups beyond finite cases, identifying specific structural limitations.

Findings

Transitive extensions exist for edge-colored k-hypergraphs only when the number of colors is a power of two.

Transitive extensions exist for k-hypertournaments only when k is even.

The study extends the theory of transitive extensions to infinite permutation groups associated with 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.