Abstract

Schur’s Theorem (1916) states that for every finite coloring of $\N$ there exists a monochromatic triple $a, b, a+b$. Several decades later, Folkman extended this statement by including in a same color arbitrarily long sequences and all finite sums from them. A breakthrough was made in 1974 by Hindman, who showed, in the same setting, the existence of an infinite sequence such that all finite sums are monochromatic, and one year later the theorem was extended to all associative operations. In this talk we explore the case of exponentiation, first investigated by Sisto (2011) and recently by Sahasrabudhe (2018). The latter proved a oFolkman Theorem for product and exponentiation at the same time. In our main theorem we realise for exponentiation the passage from finite to infinite made by Hindman for sums, by showing that for every finite coloring of $\N$ there exists an infinite sequence such that all finite exponentiations are monochromatic. We also extend the theorem to a larger class of binary non-associative operations which somehow behave in the same manner as exponentiation. This is joint work with Mauro Di Nasso.