A combinatorial theorem in group theory
Author:
E. G. Straus
Journal:
Math. Comp. 29 (1975), 303309
MSC:
Primary 20F10; Secondary 05C15
MathSciNet review:
0367072
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Abstract: There is an antiRamsey 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 kcoloring of A so that the inhomogeneous equation has no solutions with for all . Here the number of colors k can be chosen bounded by which depends on n alone and not on the or b. For nonAbelian groups an analogous qualitative result is proven when b is "residually compact". Applications to antiRamsey 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
(47 #4825)
 [2]
R.
Rado, Note on combinatorial analysis, Proc. London Math. Soc.
(2) 48 (1943), 122–160. MR 0009007
(5,87a)
 [1]
 P. ERDÖS, R. L. GRAHAM, P. MONTGOMERY, B. L. ROTHSCHILD, J. SPENCER & E. G. STRAUS, "Euclidean Ramsey theorems. I," J. Combinatorial Theory Ser. A, v. 14, 1973, pp. 341363. MR 47 #4825. MR 0316277 (47:4825)
 [2]
 R. RADO, "Note on combinatorial analysis," Proc. London Math. Soc. (2) v. 48, 1943, pp. 122160. MR 5, 87. MR 0009007 (5:87a)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197503670728
PII:
S 00255718(1975)03670728
Keywords:
Group,
Euclidean Ramsey Theorem,
residually compact
Article copyright:
© Copyright 1975
American Mathematical Society
