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An application of Ramsey’s theory to partitions in groups. I. (English) Zbl 0724.05071
In 1916 I. Schur proved the following Theorem: In every finite colouring of the positive integers N there exists a monochrome solution to the equation \(x+y=z\). The authors of this paper prove a version of Schur’s Theorem for arbitrary groups. For the equation (*) \(xy=z\), where x, y, and z are distinct non-identity elements, they obtain
Theorem A: For any n-colouring of an infinite group there exists a monochrome solution to (*).
Theorem B: For any n-colouring of a finite group of order at least \(R(2,8(n^ 2-n)/2)+1\) there exists a monochrome solution to (*). (Here the numbers R(a,b,c) are the Ramsey numbers.)
In the special cases \(n=2\) and \(n=3\), using Theorem A they obtain
Theorem C: a) If G is a 2-coloured group of order greater than 7 which is not elementary Abelian of order 9 then there is a monochrome solution of (*). b) If G is a 3-coloured group of order 17 or greater than 18 then there is a monochrome solution of (*). The proof of Theorem C: b) needs the help of a computer.

MSC:
05D10 Ramsey theory
05A17 Combinatorial aspects of partitions of integers
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References:
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