Structured sunflowers and canonical Ramsey properties

Abstract

A first-order structure $M$ is said to have the infinite sunflower property if, for each $k \in \mathbb{N}_+$ and each structure $M' \cong M$ whose elements are $k$-sets, there is $S \subseteq M'$, $S \cong M$, such that $S$ is a sunflower: a collection of sets such that each pair of elements has the same intersection. A class $\mathcal{K}$ of finite structures is said to have the finite sunflower property if for all $k \in \mathbb{N}_+$ and $B \in \mathcal{K}$, there is $C \in \mathcal{K}$ such that any structure $C' \cong C$ whose elements consist of $k$-sets contains a copy of $B$ which is a sunflower. These two notions were introduced by Ackerman, Karker and Mirabi in a recent paper, and give a structural generalisation of the well-known Erd\H{o}s-Rado sunflower lemma for sets. We show two results for countable ultrahomogeneous relational structures with strong amalgamation: first, the infinite sunflower property is equivalent to the canonical infinite point-Ramsey property; second, a certain strengthening of the canonical finite point-Ramsey property implies the finite sunflower property. (Here, "canonical" refers to statements analogous to the Erd\H{o}s-Rado canonical Ramsey theorem, involving colourings with infinitely many colours.) We also show that all free amalgamation classes with a single vertex isomorphism-type have the finite sunflower property, as do many classes of finite metric spaces, and we give a variety of further examples and observations.