This paper is dedicated to the memory of András Hajnal (1931-2016) Abstract. We show that various analogs of Hindman’s theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring c: R → Q, such that for every X ⊆ R with |X| = |R|, and every colour γ ∈ Q, there are two distinct elements x0, x1 of X for which c(x0 + x1) = γ. This forms a simultaneous generalization of a theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah. Theorem 2. For every abelian group G, there exists a colouring c: G → Q such that for every uncountable X ⊆ G and every colour γ, for some large enough integer n, there are pairwise distinct elements x0,…, xn of X such that c(x0 + ・ ・ ・ + xn) = γ. In addition, it is consistent that the preceding statement remains valid even after enlarging the set of colours from Q to R. Theorem 3. Let _κ assert that for every abelian group G of cardinality κ, there exists a colouring c: G → G such that for every positive integer n, every X0,…,Xn ∈ [G]κ, and every γ ∈ G, there are x0 ∈ X0,…, xn ∈ Xn such that c(x0 + ・ ・ ・ + xn) = γ. Then (Formula Found) holds for unboundedly many uncountable cardinals κ, and it is consistent that (Formula Found) holds for all regular uncountable cardinals κ.
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Received by the editors September 23, 2016 and, in revised form, November 14, 2016. 2010 Mathematics Subject Classification. Primary 03E02; Secondary 03E75, 03E35. Key words and phrases. Hindman’s Theorem, commutative cancellative semigroups, strong coloring, Jónsson cardinal. The first author was partially supported by Postdoctoral Fellowship number 263820/275049 from the Consejo Nacional de Ciencia y Tecnología (CONACyT), Mexico. The second author was partially supported by the Israel Science Foundation (grant #1630/14).
© 2017 American Mathematical Society.
- Commutative cancellative semigroups
- Hindman’s theorem
- JÓnsson cardinal
- Strong coloring