Strong failures of higher analogs of Hindman’s theorem

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,…,X_n ∈ [G]^κ, and every γ ∈ G, there are x₀ ∈ X₀,…, x_n ∈ X_n such that c(x₀ + ・ ・ ・ + x_n) = 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.