Let B be a finite structure and let A be its substructure. We say that B is an EPPA-witness for A if every isomorphism between substructures of A extends to an automorphism of B. A class of finite structures has the extension property for partial automorphisms (EPPA, also called the Hrushovski property) if every member of the class has an EPPA-witness in the class. In 1992, Hrushovski proved that the class of finite graphs has EPPA, a key ingredient in the subsequent proof of the small index property for the random graph by Hodges, Hodkinson, Lascar, and Shelah.

It turns out that under natural minor assumptions, a class with EPPA has the amalgamation property, and hence has a Fraisse limit F. EPPA is then equivalent to Aut(F) being the closure of a chain of compact subgroups of Aut(F). In 2005, Bhattacharjee and Macpherson proved that one can construct EPPA-witnesses for graphs which respect compositions of partial automorphisms, and that this implies that the automorphism group of the random graph contains a dense locally finite subgroup. Solecki and Siniora later generalised their methods, formalised the notion of coherent EPPA and proved that the, at the time, strongest sufficient condition for a class to have EPPA actually implies coherent EPPA. This was further strengthened by Hubička, Konečný, and Nešetřil in 2019.

Nevertheless, EPPA has recently been proved for a few classes using ad hoc methods which turned out not to give coherent EPPA. In one case (two-graphs) it was possible to obtain a dense locally finite subgroup by other means, but in the other two cases (n-partite tournaments and semigeneric tournaments), both coherent EPPA and the existence of a dense locally finite subgroup remain open.

Despite a fancy-sounding title, the talk, which is based on joint work with (subsets of) Evans, Hubička, Jahel, Nešetřil, and Sabok, will be pretty basic.