# Ramsey properties and extending partial automorphisms for classes of finite structures

## TL;DR

This paper proves that free amalgamation classes of finite structures with relations and partial functions become Ramsey classes when ordered linearly, and establishes extension properties for automorphisms, solving multiple conjectures.

## Contribution

It extends the Nešetřil-Rödl Theorem to classes with functions and relations, and introduces new criteria for Ramsey properties and automorphism extensions.

## Key findings

- Free amalgamation classes are Ramsey when ordered linearly.
- Subclasses with the ordering property are identified.
- Extension property for partial automorphisms is established for certain classes.

## Abstract

We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the Ne\v{s}et\v{r}il-R\"odl Theorem and the second and third authors' Ramsey theorem for finite models (that is, structures with both relations and functions). We also find subclasses with the ordering property. For languages with relational symbols and unary functions we also show the extension property for partial automorphisms (EPPA) of free amalgamation classes. These general results solve several conjectures and provide an easy Ramseyness test for many classes of structures.