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.

Extending partial isometries of antipodal graphs

“…Then for every A ∈ A δ K there is B ∈ A δ K which is an EPPA-witness for A. Proposition 3.1 extends the results of [9] where it was proved for diameter 3.…”

“…In this paper we combine the results of [4] with the ideas from [9] and the new strengthening of the Herwig-Lascar theorem by Hubička, Nešetřil and the author [22] (stated here in a weaker form as Theorem 2.5) and prove the following theorem, thereby answering a question of Aranda et al (Problem 1.3 in [4]) and completing the study of EPPA for classes from Cherlin's list. Theorem 1.2.…”

“…class of antipodal metric spaces of diameter 3 is closely connected to switching classes of graphs and to two-graphs (see [9]). The following fact summarizes results from [4] about the non-bipartite odd diameter antipodal classes.…”

“…Hrushovski's proof was group-theoretical, Herwig and Lascar [15] later gave a simple combinatorial proof by embedding G into the complement of a Kneser graph. After this, the quest of identifying new classes of structures with EPPA continued with a series of papers including [4,8,9,13,14,15,17,20,21,22,25,30,33,34].…”

“…EPPA and other combinatorial properties of classes from Cherlin's list were studied by Aranda, Bradley-Williams, Hng, Hubička, Karamanlis, Kompatscher, Pawliuk and the author [2,3,4] (see also [24]) and in [4] almost all the questions were settled, only EPPA for antipodal classes of odd diameter and bipartite antipodal classes of even diameter (see Section 2.3) remained open. An important step was later done by Evans, Hubička, Nešetřil and the author [9] who proved EPPA for antipodal metric spaces of diameter 3.…”