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EPPA for two-graphs and antipodal metric spaces


Authors: David M. Evans, Jan Hubička, Matěj Konečný and Jaroslav Nešetřil
Journal: Proc. Amer. Math. Soc. 148 (2020), 1901-1915
MSC (2010): Primary 05E18, 20B25, 22F50, 03C15, 03C52
DOI: https://doi.org/10.1090/proc/14872
Published electronically: January 15, 2020
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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.


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Additional Information

David M. Evans
Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
Email: david.evans@imperial.ac.uk

Jan Hubička
Affiliation: Faculty of Mathematics and Physics, Department of Applied Mathematics (KAM), Charles University, Prague, Czech Republic
Email: hubicka@kam.mff.cuni.cz

Matěj Konečný
Affiliation: Faculty of Mathematics and Physics, Department of Applied Mathematics (KAM), Charles University, Prague, Czech Republic
Email: matej@kam.mff.cuni.cz

Jaroslav Nešetřil
Affiliation: Faculty of Mathematics and Physics, Computer Science Institute of Charles University (IUUK), Charles University, Prague, Czech Republic
Email: nesetril@kam.mff.cuni.cz

DOI: https://doi.org/10.1090/proc/14872
Received by editor(s): February 20, 2019
Received by editor(s) in revised form: September 11, 2019
Published electronically: January 15, 2020
Additional Notes: The last three authors were supported by ERC Synergy grant DYNASNET 810115
The second and third authors were supported by project 18-13685Y of the Czech Science Foundation (GAČR)
The third author was also supported by the Charles University Grant Agency (GA UK), project 378119
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2020 American Mathematical Society