Toward an uncountable analogue of Gallai’s theorem for colorings of the plane. (English) Zbl 1393.05102
Summary: 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 N. Brown et al. [Geombinatorics 3, No. 2, 24–31 (1993; Zbl 0846.05030)] on 2-colorings of \(\mathbb E^2\) to any finite coloring of \(\mathbb E^2\).
