Abstract
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.
Citation
Peter Cameron. Javier Cilleruelo. Oriol Serra. "On monochromatic solutions of equations in groups." Rev. Mat. Iberoamericana 23 (1) 385 - 395, April, 2007.
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