Monochromatic exponential triples: an ultrafilter proof

A cornerstone of Arithmetic Ramsey Theory is \emph{Hindman's Theorem} of 1974: ``For every finite coloring of the natural numbers, there exists an infinite sequence $(x_n)$ such that all finite sums $x_{n_1}+\ldots+x_{n_k}$ of distinct elements are monochromatic".

We extend the validity of Hindman's theorem to a broad class of non-associative operations that generalize exponentiation between natural numbers.

The main tool we use in our proofs is given by the central sets, a special class of sets isolated in 1981 by H.

Furstenberg in the context of topological dynamics.

It was later discovered in 1990 by V.

Bergelson and N.

Hindman that central sets can be characterized as those sets that belong to a minimal idempotent ultrafilter, thus opening up the study of their rich combinatorial structure with the well-developed machinery of algebra in the space of ultrafilters.

As a corollary of our main result, we obtain an extension of Sahasrabudhe's results of 2018 about the existence of arbitrarily large (but finite) monochromatic exponential patterns; indeed, we obtain the existence of an infinite sequence such that all finite exponential configurations originating from its elements are monochromatic.