# All those EPPA classes (Strengthenings of the Herwig-Lascar theorem)

## TL;DR

This paper proves a comprehensive and elementary extension property for partial automorphisms (EPPA) for classes of structures with relations, unary functions, and permutation groups, unifying and strengthening several key theorems.

## Contribution

It provides a unified, combinatorial proof of EPPA for diverse classes, extending previous theorems and establishing a common framework for EPPA and Ramsey properties.

## Key findings

- Unified proof of EPPA for classes with relations and unary functions
- Extension of EPPA results to classes with forbidden homomorphisms
- Application to a problem from Hrushovski's construction

## Abstract

In this paper we prove a general theorem showing the extension property for partial automorphisms (EPPA, also called the Hrushovski property) for classes of structures containing relations and unary functions, optionally equipped with a permutation group of the language. The proof is elementary, combinatorial and fully self-contained. Our result is a common strengthening of the Herwig-Lascar theorem on EPPA for relational classes with forbidden homomorphisms, the Hodkinson-Otto theorem on EPPA for relational free amalgamation classes, its strengthening for unary functions by Evans, Hubi\v{c}ka and Ne\v{s}et\v{r}il and their coherent variants by Siniora and Solecki. We also prove an EPPA analogue of the main results of J. Hubi\v{c}ka and J. Ne\v{s}et\v{r}il: All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms), thereby establishing a common framework for proving EPPA and the Ramsey property. Our results have numerous applications, we include a solution of a problem related to a class constructed by the Hrushovski predimension construction.