## Logic Seminar

This is based on joint work with David Evans, Jan Hubička and Jaroslav Nešetřil. The extension property for partial automorphisms (EPPA), also called the Hrushovski property, is a property of classes of finite structures stating that for every $A$ there is $B$ containing $A$ as a substructure such that every isomorphism of substructures of $A$ extends to an automorphism of $B$. Every class with EPPA is an amalgamation class, in fact, EPPA is equivalent to some properties of the automorphism group of the Fraisse limit of the class. In particular, EPPA is a key ingredient in proving ample generics, the small index property etc. In this talk, we show a new easy way of proving EPPA for the class of all finite graphs and then explain how to extend these techniques to get the strongest sufficient condition for EPPA so far, in particular strengthening the theorems of Herwig and Lascar, Siniora and Solecki and Hodkinson and Otto. The talk will be self-contained.