Monochromatic exponential triples: an ultrafilter proof

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.