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A Combinatorial Theorem in Group Theory

E. G. Straus
Mathematics of Computation
Vol. 29, No. 129 (Jan., 1975), pp. 303-309
DOI: 10.2307/2005482
Stable URL: http://www.jstor.org/stable/2005482
Page Count: 7
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A Combinatorial Theorem in Group Theory
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Abstract

There is an anti-Ramsey theorem for inhomogeneous linear equations over a field, which is essentially due to R. Rado [2]. This theorem is generalized to groups to get sharper quantitative and qualitiative results. For example, it is shown that for any Abelian group $A$ (written additively) and any mappings $f_1,\cdots, f_n$ of $A$ into itself there exists a $k$-coloring $\chi$ of $A$ so that the inhomogeneous equation $$\sum^n_{i=1} (f_i(x_i) - f_i(y_i)) = b,\quad b \neq 0$$ has no solutions $x_i, y_i$ with $\chi(x_i) = \chi(y_i)$ for all $i = 1, \cdots, n$. Here the number of colors $k$ can be chosen bounded by $(3n)^{n-1}$ which depends on $n$ alone and not on the $f_i$ or $b$. For non-Abelian groups an analogous qualitative result is proven when $b$ is "residually compact". Applications to anti-Ramsey results in Euclidean geometry are given.

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