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 of *A* into itself there exists a *k*-coloring of *A* so that the inhomogeneous equation

*k*can be chosen bounded by which depends on

*n*alone and not on the 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.

**[1]**P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus,*Euclidean Ramsey theorems. I*, J. Combinatorial Theory Ser. A**14**(1973), 341–363. MR**0316277****[2]**R. Rado,*Note on combinatorial analysis*, Proc. London Math. Soc. (2)**48**(1943), 122–160. MR**0009007**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1975-0367072-8

Keywords:
Group,
Euclidean Ramsey Theorem,
residually compact

Article copyright:
© Copyright 1975
American Mathematical Society