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Density versions of Schur’s theorem for ideals generated by submeasures. (English) Zbl 1230.05036
Summary: We characterize ideals of subsets of natural numbers for which some versions of Schur’s theorem hold. These are similar to generalizations shown by {\it V. Bergelson} in [J. Comb. Theory, Ser. A 43, 338--343 (1986; Zbl 0607.10040)] and {\it P. Frankl}, {\it R. I. Graham} and {\it V. Rödl} in [J. Comb. Theory, Ser. A 54, No. 1, 95--111 (1990; Zbl 0738.05008)]. Additionally, we prove a generalization of an iterated version of Ramsey’s theorem.

05A17Partitions of integers (combinatorics)
Full Text: DOI
[1] Bergelson, Vitaly: A density statement generalizing Schur’s theorem, J. combin. Theory ser. A 43, No. 2, 338-343 (1986) · Zbl 0607.10040 · doi:10.1016/0097-3165(86)90074-9
[2] Bergelson, Vitaly; Hindman, Neil: Density versions of two generalizations of Schur’s theorem, J. combin. Theory ser. A 48, No. 1, 32-38 (1988) · Zbl 0642.05002 · doi:10.1016/0097-3165(88)90072-6
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[7] Frankl, P.; Graham, R. L.; Rödl, V.: Iterated combinatorial density theorems, J. combin. Theory ser. A 54, No. 1, 95-111 (1990) · Zbl 0738.05008 · doi:10.1016/0097-3165(90)90008-K
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[12] Solecki, Sławomir: Analytic ideals and their applications, Ann. pure appl. Logic 99, No. 1 -- 3, 51-72 (1999) · Zbl 0932.03060 · doi:10.1016/S0168-0072(98)00051-7
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