Finding unavoidable colorful patterns in multicolored graphs

TL;DR

This paper generalizes a Ramsey-type problem about multicolored complete graphs, providing characterizations and bounds for unavoidable colorful patterns in well-balanced colorings, extending previous results to multiple colors and infinite settings.

Contribution

It offers a complete characterization and asymptotically tight bounds for the minimal size ensuring unavoidable patterns in multicolored graphs, generalizing prior bipartite results to arbitrary colors and infinite spaces.

Findings

Characterization of graphs with finite Ramsey numbers for any number of colors.

Asymptotically tight bounds for these Ramsey numbers.

Extension of results to graphs on Polish spaces with non-meagre color classes.

Abstract

We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollob\'as, concerning colorings of $K_n$ where each color is well-represented. Let $\chi$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $\chi$ $\varepsilon$-balanced if all color classes have $\varepsilon$ fraction of the edges. Fix some graph $H$, together with an $r$-coloring of its edges. Consider the smallest natural number $R_\varepsilon^r(H)$ such that for all $n\geq R_\varepsilon^r(H)$, all $\varepsilon$-balanced colorings $\chi$ of $K_n$ contain a subgraph isomorphic to $H$ in its coloring. Bollob\'as conjectured a simple characterization of $H$ for which $R_\varepsilon^2(H)$ is finite, which was later proved by Cutler and Mont\'agh. Here, we obtain a characterization for arbitrary values of $r$, as well as asymptotically tight bounds. We also discuss generalizations to graphs defined on perfect Polish spaces, where the corresponding notion of balancedness is each color class being non-meagre.