An infinite cardinal version of Gallai's Theorem for colorings of the plane

TL;DR

This paper extends Gallai's theorem to infinite cardinals, showing that in any finite coloring of the plane, there are continuum many monochromatic homothetic copies of any finite point set, with a similar result in higher dimensions.

Contribution

It generalizes Gallai's theorem to infinite cardinalities and higher-dimensional Euclidean spaces, demonstrating the existence of uncountably many monochromatic homothetic copies.

Findings

Existence of 2^{}_0 monochromatic homothetic copies in the plane.

Generalization to n-dimensional Euclidean spaces.

Strengthening of Gallai's original finite set result.

Abstract

We generalize a result of Tibor Gallai as follows: for any finite set of points $\mathcal{S}$ in the plane, if the plane is colored in finitely many colors, then there exist $2^{\aleph_0}$ monochromatic subsets of the plane homothetic to $\mathcal{S}$. Furthermore, we prove an even stronger result for $n$-dimensional Euclidean space.