Strong Failures of Higher Analogs of Hindman’s Theorem

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 ~κ holds for unboundedly many uncountable cardinals κ, and it is consistent that ~κ holds for all regular uncountable cardinals κ.