Revista Matemática Iberoamericana

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Volume 23, Issue 1, 2007, pp. 385–395
DOI: 10.4171/RMI/499

Published online: 2007-04-30

On monochromatic solutions of equations in groups

Peter J. Cameron[1], Javier Cilleruelo[2] and Oriol Serra[3]

(1) Queen Mary University of London, UK
(2) Universidad Autónoma de Madrid, Spain
(3) Universitat Politècnica de Catalunya, Barcelona, Spain

We show that the number of monochromatic solutions of the equation $x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_r^{\alpha_r}=g$ in a $2$-coloring of a finite group $G$, where $\alpha_1,\ldots,\alpha_r$ are permutations and $g\in G$, depends only on the cardinalities of the chromatic classes but not on their distribution. We give some applications to arithmetic Ramsey statements.

Keywords: Orthogonal arrays, Schur triples, monochromatic arithmetic progressions

Cameron Peter, Cilleruelo Javier, Serra Oriol: On monochromatic solutions of equations in groups. Rev. Mat. Iberoamericana 23 (2007), 385-395. doi: 10.4171/RMI/499