Sunflowerable structures

Abstract

We call an infinite structure $\mathcal{M}$ <em>sunflowerable</em> if whenever $\mathcal{M}'$ is isomorphic to $\mathcal{M}$ with underlying set $\mathcal{M}'$, consisting of finite sets of bounded size, there is an $M_0 \subseteq M'$ such that $M_0$ is a sunflower and $\mathcal{M}'{\restriction_{M_0}}$ is isomorphic to $\mathcal{M}$. We give sufficient conditions on $\mathcal{M}$ to show that $\mathcal{M}$ is sunflowerable. These conditions allow us to show that several well-known structures are sunflowerable and give a complete characterization of the countable linear orderings that are sunflowerable.

We show that a sunflowerable structure must be indivisible. This allows us to show that any Fraïssé limit that has the 3-disjoint amalgamation property and a single unary type must be indivisible.

In addition to studying sunflowerability of infinite structures we also consider an analogous property of an age, which we call the <em>sunflower property</em>. We show that any sunflowerable structure must have an age with the sunflower property. We also give concrete bounds in the case that the age has the hereditary property, the 3-disjoint amalgamation property and is indivisible.