Additive and multiplicative Ramsey theory in the reals and the rationals. (English) Zbl 0917.05080

Bergelson, Vitaly; Hindman, Neil; Leader, Imre

J. Comb. Theory, Ser. A 85, No. 1, 41-68 (1999).

The paper discusses Ramsey theoretic results for finite partitions of the real interval \((0,1)\). Using two different approaches, one based on Stone-Čech compactifications and the other based on elementary methods, the authors show that for every finite partition of the real interval \((0,1)\) into measurable subsets or into Baire sets, respectively, at least one member of the partition contains a sequence with all of its finite sums and products. Moreover, partitions of the rationals and of the dyadic rationals are discussed, for which separate additive and multiplicative statements are derived. A counterexample is given showing that even weak versions of the combined additive and multiplicative results that were proven for the real interval \((0,1)\) do not hold for the dyadic rationals. The related question for the rationals remains open, but a negative answer is conjectured also for this case.

Reviewer: Kathrin Klamroth (Kaiserslautern)

MSC:

05D10

Ramsey theory

Keywords:

additive Ramsey theory; multiplicative Ramsey theory

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