Thick, syndetic, and piecewise syndetic subsets of Fra\"iss\'e structures

TL;DR

This paper systematically studies thick, syndetic, and piecewise syndetic subsets within Fra"iss"e structures, exploring their properties, ultrafilters, and generalizing previous results in the area.

Contribution

It introduces a unified framework for analyzing these subsets in Fra"iss"e structures and extends existing results on ultrafilters and combinatorial properties.

Findings

Defined and studied ultrafilters on these subsets.

Generalized results of Bergelson, Hindman, and McCutcheon.

Explored abstract questions related to families of sets.

Abstract

We define and undertake a systematic study of thick, syndetic, and piecewise syndetic subsets of a Fra\"iss\'e structure. Each of these collections forms a family in the sense of Akin and Glasner [AG], and we define and study ultrafilters on each of these families, paying special attention to ultrafilters on the thick sets. In the process, we generalize many results of Bergelson, Hindman, and McCutcheon [BHM]. We also discuss some abstract questions about families implicit in the work of Akin and Glasner.