EPPA for two-graphs and antipodal metric spaces

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.

arXiv comment14 pages, 3 figures

References in corpus (4)

• All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)
• Extending partial isometries of antipodal graphs
• A combinatorial proof of the extension property for partial isometries
• Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of the Urysohn space

Cited by in corpus (6)

• Ramsey expansions of metrically homogeneous graphs
• Simplicity of the automorphism groups of generalised metric spaces
• EPPA numbers of graphs
• Twenty years of Nešetřil's classification programme of Ramsey classes
• Extension property for partial automorphisms of the $n$-partite and semigeneric tournaments
• Ramsey expansions of 3-hypertournaments