The Adjacent Hindman's Theorem for uncountable groups

Recent 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 Komjáth, 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 analogous 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 κ there is a λ such that, whenever a group G of cardinality λ is coloured with κ colours, there exists a λ-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 λ as a function of κ, 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.