The girth Ramsey theorem

TL;DR

This paper proves the existence of hypergraphs with large girth that satisfy Ramsey properties for given hypergraphs and colorings, including constructions that locally resemble forests of copies of the original hypergraph.

Contribution

It establishes the girth Ramsey theorem for hypergraphs, demonstrating the existence of hypergraphs with specified girth and Ramsey properties, including local forest-like structures.

Findings

Existence of hypergraphs with large girth satisfying Ramsey properties

Construction of hypergraphs locally resembling forests of a given hypergraph

Generalization of Ramsey theory to hypergraphs with girth constraints

Abstract

Given a hypergraph $F$ and a number of colours $r$, there exists a hypergraph $H$ of the same girth satisfying $H\longrightarrow (F)_r$. Moreover, for every linear hypergraph $F$ there exists a Ramsey hypergraph $H$ that locally looks like a forest of copies of $F$.