# Planar Ramsey graphs

## TL;DR

This paper investigates which graphs are guaranteed to appear monochromatically in any edge coloring of some planar graph, establishing classes of unavoidable graphs and exploring the concept in multiple colors.

## Contribution

It characterizes planar unavoidable graphs, proves certain cycles, paths, and small-radius trees are unavoidable, and introduces the concept for multiple colors.

## Key findings

- Cycles of 4 vertices are planar unavoidable.
- Paths are planar unavoidable.
- All trees with radius at most 2 are planar unavoidable.

## Abstract

We say that a graph $H$ is planar unavoidable if there is a planar graph $G$ such that any red/blue coloring of the edges of $G$ contains a monochromatic copy of $H$, otherwise we say that $H$ is planar avoidable. I.e., $H$ is planar unavoidable if there is a Ramsey graph for $H$ that is planar. It follows from the Four-Color Theorem and a result of Gon\c{c}alves that if a graph is planar unavoidable then it is bipartite and outerplanar. We prove that the cycle on $4$ vertices and any path are planar unavoidable. In addition, we prove that all trees of radius at most $2$ are planar unavoidable and there are trees of radius $3$ that are planar avoidable. We also address the planar unavoidable notion in more than two colors.