# EPPA for two-graphs and antipodal metric spaces

## TL;DR

This paper proves that finite two-graphs and certain antipodal metric spaces have the extension property for partial automorphisms, providing new insights into their automorphism structures and answering longstanding questions.

## Contribution

It establishes EPPA for finite two-graphs and antipodal metric spaces using a combinatorial proof, addressing open problems and highlighting unique behaviors of two-graphs.

## Key findings

- EPPA holds for finite two-graphs
- EPPA holds for integer valued antipodal metric spaces of diameter 3
- Two-graphs exhibit unique properties differing from other classes with EPPA

## Abstract

We prove that the class of finite two-graphs has the extension property for partial automorphisms (EPPA, or Hrushovski property), thereby answering a question of Macpherson. In other words, we show that the class of graphs has the extension property for switching automorphisms. We present a short, self-contained, purely combinatorial proof which also proves EPPA for the class of integer valued antipodal metric spaces of diameter 3, answering a question of Aranda et al. The class of two-graphs is an important new example which behaves differently from all the other known classes with EPPA: Two-graphs do not have the amalgamation property with automorphisms (APA), their Ramsey expansion has to add a graph, it is not known if they have coherent EPPA and even EPPA itself cannot be proved using the Herwig--Lascar theorem.