Thick, syndetic, and piecewise syndetic subsets of Fraïssé structures

We define and undertake a systematic study of thick, syndetic, and piecewise syndetic subsets of a Fraïssé structure. Each of these collections forms a family in the sense of Akin and Glasner [1], 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 [4]. We also discuss some abstract questions about families implicit in the work of Akin and Glasner.