Thick, syndetic, and piecewise syndetic subsets of Fraïssé structures (Q526787)

Let \(G\) be a discrete group, and let \(\mathrm{Fin}(G)\) denote the finite subsets of \(G\). In [Proceedings of the 1998 topology and dynamics conference. Vol. 23 (Spring). Fairfax, VA: George Mason University (1998; Zbl 0944.00064)] the following definitions are given: (i) \(T \subseteq G\) is thick if for every \(E \in \mathrm{Fin}(G)\), there is \(g \in G\) with \(gE \subseteq T\). (ii) \(S \subseteq G\) is syndetic if \(G \setminus S\) is not thick, (iii) \(P \subseteq G\) is piecewise syndetic if there is \(E \in \mathrm{Fin}(G)\) with \(\cup_{g\in E} Pg^{-1}\) thick. Now, if \(L\) is a relational language, a Fraïssé class \(\mathcal{K}\) is a class of \(L\)-structures with the following four properties: 1. \(\mathcal{K}\) contains only finite structures, contains structures of arbitrarily large finite cardinality, is closed under isomorphism, and contains only countably many isomorphism classes. 2. \(\mathcal{K}\) has the hereditary property: if \(B\in \mathcal{K}\) and \(A \subseteq B\), then \(A \in \mathcal{K}\). 3. \(\mathcal{K}\) has the joint embedding property: if \(A\), \(B \in \mathcal{K}\), then there is \(C \in \mathcal{K}\) which embeds both \(A\) and \(B\). 4. \(\mathcal{K}\) has the amalgamation property: if \(A\), \(B\), \(C \in \mathcal{K}\) and \(f : A \to B\) and \(g : A \to C\) are embeddings, there is \(D \in \mathcal{K}\) and embeddings \(r : B \to D\) and \(s : C \to D\) with \(r \circ f = s \circ g\). The purpose of this paper is to undertake a systematic study of thick, syndetic, and piecewise syndetic subsets of a Fraïssé structure.