Universal flows of closed subgroups of $S_{\infty}$ and relative extreme amenability

TL;DR

This paper explores the universality of continuous actions linked to pairs of closed subgroups of $S_{ ext{infty}}$, introducing new concepts like relative extreme amenability and relative Ramsey properties to deepen understanding.

Contribution

It introduces three new concepts—relative extreme amenability, and two forms of relative Ramsey properties—that advance the understanding of universality in group actions.

Findings

Identifies the relevance of new concepts in understanding universality.

Provides partial answers to longstanding questions in the field.

Connects group properties with combinatorial Ramsey properties.

Abstract

This paper is devoted to the study of universality for a particular continuous action naturally attached to certain pairs of closed subgroups of $S_{\infty}$. It shows that three new concepts, respectively called relative extreme amenability, relative Ramsey property for embeddings, and relative Ramsey property for structures, are relevant in order to understand this property correctly. It also allows to provide a partial answer to a question posed by Kechris, Pestov and Todorcevic.