Monochromatic exponential triples: an ultrafilter proof

TL;DR

This paper provides a new ultrafilter-based proof for the existence of monochromatic exponential triples in finite colorings of natural numbers, and extends the method to infinite patterns.

Contribution

It introduces a novel ultrafilter proof technique for exponential triples and generalizes it to infinite monochromatic exponential patterns.

Findings

Existence of monochromatic exponential triples proved using ultrafilters

Generalization to infinite monochromatic exponential patterns

New ultrafilter construction for exponential pattern existence

Abstract

We present a short ultrafilter proof of the existence of monochromatic exponential triples $\{a, b, b^a\}$ in any finite coloring of the natural numbers. The proof is given from scratch and uses only Ramsey's theorem, the notion of asymptotic density and the definition of ultrafilter as prerequisites. We then generalize the construction using a special ultrafilter whose existence is well known in the algebra of ultrafilters, and prove a new result on the existence of infinite monochromatic exponential patterns.