On globally sparse Ramsey graphs

TL;DR

This paper investigates the minimal density of globally sparse graphs that guarantee a monochromatic copy of certain graphs under edge colorings, extending known results beyond cliques to bipartite graphs, cycles, and paths.

Contribution

It determines the Ramsey density for various graphs like bipartite graphs, cycles, and paths, advancing understanding of sparse Ramsey graphs beyond previous clique-focused results.

Findings

Ramsey density is characterized for complete bipartite graphs.

Results include bounds for cycles and paths.

The work extends sparse Ramsey graph theory to new graph classes.

Abstract

We say that a graph $G$ has the Ramsey property w.r.t.\ some graph $F$ and some integer $r\geq 2$, or $G$ is $(F,r)$-Ramsey for short, if any $r$-coloring of the edges of $G$ contains a monochromatic copy of $F$. R{\"o}dl and Ruci{\'n}ski asked how globally sparse $(F,r)$-Ramsey graphs $G$ can possibly be, where the density of $G$ is measured by the subgraph $H\subseteq G$ with the highest average degree. So far, this so-called Ramsey density is known only for cliques and some trivial graphs $F$. In this work we determine the Ramsey density up to some small error terms for several cases when $F$ is a complete bipartite graph, a cycle or a path, and $r\geq 2$ colors are available.