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Mathematics of Computation

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A combinatorial theorem in group theory

Author: E. G. Straus
Journal: Math. Comp. 29 (1975), 303-309
MSC: Primary 20F10; Secondary 05C15
MathSciNet review: 0367072
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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 qualitative 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

$\displaystyle \sum\limits_{i = 1}^n {({f_i}({x_i}) - {f_i}({y_i})) = b,\quad b \ne 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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Keywords: Group, Euclidean Ramsey Theorem, residually compact
Article copyright: © Copyright 1975 American Mathematical Society

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