Ramsey-type numbers involving graphs and hypergraphs with large girth

TL;DR

This paper investigates the minimal size of graphs with large girth that guarantee monochromatic cycles under any edge coloring, connecting Ramsey numbers, graph girth, and related combinatorial problems.

Contribution

It establishes upper bounds on the size of such graphs based on Ramsey numbers and girth, advancing understanding of Ramsey-type problems in graphs and hypergraphs.

Findings

Existence of graphs with specified girth and Ramsey properties on bounded vertices

Derived bounds involving Ramsey numbers and girth for minimal graphs

Explored related problems in arithmetic progressions and clique formations

Abstract

A question of Erd\H{o}s asks if for every pair of positive integers $r$ and $k$, there exists a graph $H$ having $\textrm{girth}(H)=k$ and the property that every $r$-colouring of the edges of $H$ yields a monochromatic cycle $C_k$. The existence of such graphs was confirmed by the third author and Ruci\'nski. We consider the related numerical problem of determining the smallest such graph with this property. We show that for integers $r$ and $k$, there exists a graph $H$ on $R^{10k^2} k^{15k^3}$ vertices (where $R = R(C_k;r)$ is the $r$-colour Ramsey number for the cycle $C_k$) having $\textrm{girth}(H)=k$ and the Ramsey property that every $r$-colouring of $E(H)$ yields a monochromatic $C_k$. Two related numerical problems regarding arithmetic progressions in sets and cliques in graphs are also considered.