The weak Ramsey property and extreme amenability

TL;DR

This paper extends the Kechris--Pestov--Todorčević correspondence to weak Fraïssé categories using the weak Ramsey property, with applications to various categories like monoids and trees.

Contribution

It introduces the weak Ramsey property into the correspondence, broadening its applicability to new categories and automorphism groups.

Findings

Established the weak Ramsey property for several categories.

Extended the correspondence to automorphism groups of generic objects.

Demonstrated applications to monoid categories, linear orders, and trees.

Abstract

We extend the Kechris--Pestov--Todor\v{c}evi\'c correspondence to weak Fra\"{\i}ss\'{e} categories and automorphism groups of generic objects. The new ingredient is the weak Ramsey property. We demonstrate the theory on several examples including monoid categories, the category of almost linear orders, and categories of strong embeddings of trees.