# Fixed points in compactifications and combinatorial counterparts

## TL;DR

This paper generalizes the Kechris-Pestov-Todorcevic correspondence, showing that fixed points in group compactifications relate to Ramsey properties, thus revealing a broader connection between dynamics and combinatorics.

## Contribution

It recasts the classical correspondence as part of a more general framework linking fixed points in compactifications to Ramsey-type combinatorial statements.

## Key findings

- Ramsey properties emerge as fixed point conditions in group compactifications
- The framework applies to various dynamical contexts beyond non-Archimedean groups
- Establishes a unified view connecting dynamics and combinatorics

## Abstract

The Kechris-Pestov-Todorcevic correspondence connects extreme amenability of non-Archimedean Polish groups with Ramsey properties of classes of finite structures. The purpose of the present paper is to recast it as one of the instances of a more general construction, allowing to show that Ramsey-type statements actually appear as natural combinatorial expressions of the existence of fixed points in certain compactifications of groups, and that similar correspondences in fact exist in various dynamical contexts.