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Monochrome symmetric subsets of colored groups. (English) Zbl 1121.20034

Summary: The first author [in Electron. J. Comb. 10, R28, 8 p. (2003); printed version J. Comb. 10, No. 3 (2003; Zbl 1023.05132)] asked three questions. Is it true that every infinite group admitting a 2-coloring without infinite monochromatic symmetric subsets is either almost cyclic (i.e., have a finite index subgroup which is cyclic infinite) or countable locally finite? Does every infinite group \(G\) include a monochromatic symmetric subset of any cardinal \(<|G|\) for any finite coloring? Does every uncountable group \(G\) such that \(|B(G)|<|G|\) where \(B(G)=\{x\in G:x^2=1\}\), admit a 2-coloring without monochromatic symmetric subsets of cardinality \(|G|\)?
We answer the first question positively. Assuming the generalized continuum hypothesis (GCH), we give a positive answer to the second question in the Abelian case. Finally, we build a counter-example for the third question and we give a necessary and sufficient condition for an infinite group \(G\) to admit 2-coloring without monochromatic symmetric subsets of cardinality \(|G|\). This generalizes some results of I. V. Protasov on infinite Abelian groups [Math. Notes 59, No. 3, 336-338 (1996); translation from Mat. Zametki 59, No. 3, 468-471 (1996; Zbl 0872.20048); Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky 1999, No. 11, 54-57 (1999; Zbl 0951.22001)].

MSC:

20F99 Special aspects of infinite or finite groups
20F50 Periodic groups; locally finite groups
20E07 Subgroup theorems; subgroup growth
20K99 Abelian groups
05D10 Ramsey theory
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References:

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[2] Gromov, M., Groups of Polynomial growth and expanding maps, Publ. Math. IHES Bures Sur Yvette, 53, 53-73 (1981) · Zbl 0474.20018
[3] Y. Gryshko, Monochrome symmetric subsets in 2-colorings of groups, Electron. J. Combin. 10 (2003); http:www.combinatorics.org/Volume_10/Abstracts/v10i1r28.html; Y. Gryshko, Monochrome symmetric subsets in 2-colorings of groups, Electron. J. Combin. 10 (2003); http:www.combinatorics.org/Volume_10/Abstracts/v10i1r28.html · Zbl 1023.05132
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[5] Protasov, I., Monochromatic symmetric subsets in colorings of abelian groups, Dopovidi NAN Ukrain., 1, 54-57 (1999) · Zbl 0951.22001
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