Structured Sunflowers

TL;DR

This paper introduces the concept of sunflowerability in infinite structures, providing conditions for it, characterizing certain structures, and exploring related properties like indivisibility and the sunflower property of ages.

Contribution

It establishes sufficient conditions for sunflowerability, characterizes countable linear orderings that are sunflowerable, and links sunflowerability to indivisibility and properties of ages.

Findings

Several well-known structures are shown to be sunflowerable.

A complete characterization of countable linear orderings that are sunflowerable.

Sunflowerable structures must be indivisible, with implications for Fra"issé limits.

Abstract

We call an infinite structure $\mathcal{M}$ sunflowerable if whenever $\mathcal{M}'$ is isomorphic to $\mathcal{M}$ with underlying set $M'$, consisting of finite sets of bounded size, there is an $M_0 \subseteq M'$ such that $M_0$ is a sunflower and $\mathcal{M}'\!\!\upharpoonright[M_0]$ 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 which are sunflowerable. We show that a sunflowerable structure must be indivisible. This allows us to show that any Fra\"iss\'e limit which 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 sunflower property. 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.