Journal of Combinatorics

Volume 5 (2014)

Number 4

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

Pages: 445 – 452

DOI: http://dx.doi.org/10.4310/JOC.2014.v5.n4.a3

Author

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

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.

Keywords

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

2010 Mathematics Subject Classification

05C50, 05D10, 52C10

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