An adjacent Hindman theorem for uncountable groups

Results of Hindman, Leader and Strauss and of the second author and Rinot showed that some natural analogs of Hindman's theorem fail for all uncountable cardinals. Results in the positive direction were obtained by Komj\'ath, the first author, and the second author and Lee, who showed that there are arbitrarily large abelian groups satisfying some Hindman-type property. Inspired by an analog result studied by the first author in the countable setting, we prove a new variant of Hindman's theorem for uncountable cardinals, called the Adjacent Hindman's Theorem: For every $\kappa$ there is a $\lambda$ such that, whenever a group $G$ of cardinality $\lambda$ is coloured in $\kappa$ colours, there exists a $\lambda$-sized injective sequence of elements of $G$ with all finite products of adjacent terms of the sequence of the same colour. We obtain bounds on $\lambda$ as a function of $\kappa$, and prove that such bounds are optimal. This is the first example of a Hindman-type result for uncountable cardinals that we can prove also in the non-Abelian setting and, furthermore, it is the first such example where monochromatic products (or sums) of unbounded length are guaranteed

Infinite Monochromatic Sumsets for Colourings of the Reals.

N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of R so that no infinite sumset X + X is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any finite colouring c of R there is an infinite X ⊆ R so that c ↾ X + X is constant