Cohen Forcing with Fraïssé Classes Beyond ${\aleph }_1$

Abstract

Kostana [5] asked for natural conditions on a class $\mathcal{K}$ to ensure that forcing with finite members of $\mathcal{K}$ on an underlying set $X$ is forcing equivalent to Cohen forcing. We show that if ${\mathcal{K}}_{△}$ is the class of finite triangle-free graphs and $|X|\ge {\aleph }_2$, then ${\mathbb{C}}_X({\mathcal{K}}_{△})\not\simeq \mathbb{C}[X].$ We then use a characterization of Cohen algebras due to Milovich to obtain a positive result. We show that if ${\aleph }_n\le |X|$ and $\mathcal{K}$ has $(n+1)$-disjoint amalgamation, then ${\mathbb{C}}_X(\mathcal{K})$ is Cohen.