Structured sunflowers and Ramsey properties
Rob Sullivan, Jeroen Winkel
TL;DR
Addresses the problem of generalizing the Erdős–Rado sunflower lemma to first-order relational structures; defines the infinite sunflower property and its local and partition-theoretic characterisations for countable Fraïssé structures $M$ with strong amalgamation, including the equivalence with the galah property and local replicability. The main results show that for relational Fraïssé structures with strong amalgamation the infinite sunflower property is equivalent to the infinite $2$-sunflower property and to the existence of monochromatic or heterochromatic copies under any colouring $\chi: M \to \omega$, and that the galah property coincides with local replicability. The paper also proves that all free amalgamation classes with a single vertex isomorphism type have the finite sunflower property via a canonical Ramsey theorem for hypergraphs, and provides extensive examples and counterexamples illustrating both the reach and limits of these properties. Together these results connect and extend AKM25's structured sunflowers, offering a framework linking indivisibility, local replicability, and Ramsey-type properties in Fraïssé theory, while raising questions about extending to more general Fraïssé structures and uncountable contexts.
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 distinct 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 $C' \cong C$ 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 Erdős-Rado sunflower lemma for sets. We characterise the infinite sunflower property for countable ultrahomogeneous relational structures with strong amalgamation in terms of a certain vertex-partition property related to indivisibility, generalising and connecting some results of Ackerman, Karker and Mirabi in the countable ultrahomogeneous context, and we show that all free amalgamation classes with a single vertex isomorphism-type have the finite sunflower property. We give a variety of examples and further observations.