Journal of Combinatorics

Volume 5 (2014)

Number 4

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

Pages: 445 – 452



Jeremy F. Alm (Department of Mathematics, Illinois College, Jacksonville, Il., U.S.A.)


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.


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

2010 Mathematics Subject Classification

05C50, 05D10, 52C10

Published 11 March 2015