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, in this note we prove a new variant of Hindman's theorem for uncountable cardinals, called Adjacent Hindman 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 $G$ such that all the finite products of adjacent terms of the sequence have the same colour. We prove optimal bounds on $\lambda$ as a function of $\kappa$. This is the first example of a Hindman-type result for uncountable cardinals that holds also in the non-abelian setting and, furthermore, it is also the first such example where monochromatic products (or sums) of arbitrary length are guaranteed.