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

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.

Keywords

Gallai’s Theorem, homothety, infinite cardinal, combinatorial geometry

2010 Mathematics Subject Classification

05C50, 05D10, 52C10

Full Text (PDF format)

Published 11 March 2015