A sunflower anti-Ramsey theorem and its applications

TL;DR

This paper extends a sunflower theorem to hypergraph edge colourings with bounded monochromatic sunflower petals, guaranteeing large rainbow subhypergraphs, with applications in geometry and algebra, including an infinite version.

Contribution

It generalizes previous sunflower theorems by incorporating bounds on monochromatic sunflower petals and applies these results to geometric and algebraic problems, also providing an infinite case.

Findings

Established a bound on monochromatic sunflower petals leading to large rainbow subhypergraphs

Extended the sunflower theorem to hypergraphs with applications in geometry and algebra

Presented an infinite version of the sunflower theorem

Abstract

A $h$-sunflower in a hypergraph is a family of edges with $h$ vertices in common. We show that if we colour the edges of a complete hypergraph in such a way that any monochromatic $h$-sunflower has at most $\lambda$ petals, then it contains a large rainbow complete subhypergraph. This extends a theorem by Lefmann, R\"odl and Wysocka, but this version can be applied to problems in geometry and algebra. We also give an infinite version of the theorem.