Extending partial automorphisms of $n$-partite tournaments

TL;DR

This paper proves that for any finite n-partite tournament, there exists a larger n-partite tournament where all partial automorphisms extend to full automorphisms, using purely combinatorial methods.

Contribution

It establishes the extension property for partial automorphisms for all finite n-partite tournaments with a combinatorial approach, extending to semi-generic tournaments.

Findings

EPPA holds for all finite n-partite tournaments

Constructs are purely combinatorial, avoiding deep group theory

Results extend to semi-generic tournaments

Abstract

We prove that for every $n\geq 2$ the class of all finite $n$-partite tournaments (orientations of complete $n$-partite graphs) has the extension property for partial automorphisms, that is, for every finite $n$-partite tournament $G$ there is a finite $n$-partite tournament $H$ such that every isomorphism of induced subgraphs of $G$ extends to an automorphism of $H$. Our constructions are purely combinatorial (whereas many earlier EPPA results use deep results from group theory) and extend to other classes such as the class of all finite semi-generic tournaments.