Toward an uncountable analogue of Gallai's Theorem for colorings of the plane

TL;DR

This paper proves that for any finite point configuration in the integer lattice, any finite coloring of the plane contains uncountably many monochromatic subsets similar to that configuration, extending previous 2-color results.

Contribution

It generalizes a known 2-coloring result to any finite coloring, establishing an uncountable abundance of monochromatic homothetic copies of finite point sets in the plane.

Findings

Uncountably many monochromatic homothetic copies exist for any finite coloring.

Extension of previous 2-color results to arbitrary finite colorings.

Generalization to all finite point configurations in the integer lattice.

Abstract

In this paper we prove that if $S$ is any finite configuration of points in $\mathbb{Z}^2$, then any finite coloring of $\mathbb{E}^2$ must contain uncountably many monochromatic subsets homothetic to $S$. We extend a result of Brown, Dunfield, and Perry on 2-colorings of $\mathbb{E}^2$ to any finite coloring of $\mathbb{E}^2$.