  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.
