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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.

MSC:

05D10 Ramsey theory
11B75 Other combinatorial number theory

Citations:

Zbl 1036.11005

Software:

GAP; RON
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Full Text: DOI EuDML

References:

[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
[2] The GAP Group: GAP -Groups, Algorithms, and Programming, Ver- sion 4.3. Aachen, St Andrews, 2002. www-gap.dcs.st-and.ac.uk/ gap.
[3] Graham, R., Rödl, V. and Ruciński, A.: On Schur properties of ran- dom subsets of integers. J. Numb. Theory 61 (1996), no. 2, 388-408. · Zbl 0880.05081 · doi:10.1006/jnth.1996.0155
[4] Schoen, T.: The Number of Monochromatic Schur Triples. European J. Combinatorics 20 (1999), no. 8, 855-866. · Zbl 0945.05062 · doi:10.1006/eujc.1999.0297
[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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