On monochromatic solutions of equations in groups. (English) Zbl 1124.05086

In his paper [Adv. Appl. Math. 31, No. 1, 193–198 (2003; Zbl 1036.11005)] B. A. Datskovsky showed that the number of monochromatic Schur triples \((x,y,z)\) with \(xy=z\) in a 2-colouring of \(\mathbb{Z}/n\mathbb{Z}\) depends only on the cardinalities of the colour classes and not on the distribution of the colours.
In the current paper a generalization to other finite groups \(G\) is given and the result then is applicated to Schur triples in \(G\), arithmetic progressions in \(G\), and Pythagorean triples in \(\mathbb{Z}/p\mathbb{Z}\). There are also some results for 3-coloured groups.


05D10 Ramsey theory
11B75 Other combinatorial number theory


Zbl 1036.11005


Full Text: DOI EuDML


[1] Datskovsky, B. A.: On the number of monochromatic Schur triples. Adv. in Appl. Math. 31 (2003), no. 1, 193-198. · Zbl 1036.11005 · doi:10.1016/S0196-8858(03)00010-1
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[5] Robertson, A. and Zeilberger, D.: A 2-coloring of [1, N ] can have (1/22)N 2 + O(N ) monochromatic Schur triples, but not less! Electron. J. Combin. 5 (1998), Research Paper 19, 4 pp. · Zbl 0894.05052
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