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

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 $\mathbb{E}^2$.