Extension property for partial automorphisms of the $n$-partite and semigeneric tournaments

TL;DR

This work proves EPPA for finite $n$-partite tournaments with $n\in\{2,3,\dots,\omega\}$ and for finite semigeneric tournaments by extending the Hrushovski-style EPPA framework to these directed graph classes. Central to the method are explicit EPPA-witness constructions $\mathbf{B}$ built from valuation functions, together with carefully designed families of automorphisms $\theta_\pi$, $\theta_{u,v}$, and, in the semigeneric case, $\theta_{a,v}$, that enable extending any partial automorphism of the base structure. The paper also establishes ample generics for the Fraïssé limit of the $\omega$-partite tournaments and, for finite $n$-partite cases, shows the lack of 1-generic automorphisms, with analogous results for semigeneric tournaments. These findings yield amenability and small index property consequences for the automorphism groups and illuminate the landscape of homogeneous directed graphs with EPPA. The results characterize how EPPA interacts with tournament-like structures and open avenues for coherent EPPA and profinite topology connections in this setting.

Abstract

We present a proof of the extension property for partial automorphisms (EPPA) for classes of finite $n$-partite tournaments for $n \in \{2,3,\ldots,ω\}$, and for the class of finite semigeneric tournaments. We also prove that the generic $ω$-partite tournament and the generic semigeneric tournament have ample generics.

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3: PawliukSokic16
• Definition 1.4: Truss1992hodges1993bKechris2007
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Lemma 2.1
• proof
• Lemma 2.2