Structured sunflowers and canonical Ramsey properties

TL;DR

This paper explores the sunflower properties in infinite and finite structures, establishing equivalences with Ramsey properties and demonstrating these properties in various classes, including free amalgamation classes and metric spaces.

Contribution

It proves the equivalence between the infinite sunflower property and the canonical infinite point-Ramsey property for certain structures, and shows many classes have the finite sunflower property.

Findings

Infinite sunflower property is equivalent to the canonical infinite point-Ramsey property.

Certain classes of structures, including free amalgamation classes with one vertex type, have the finite sunflower property.

Many classes of finite metric spaces also possess the finite sunflower property.

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.