Venue
Dipartimento di Matematica, Aula Seminari.
Abstract
Schur’s Theorem (1916) states that for every finite coloring of $\mathbb N$ there exists a monochromatic triple $a, b, a+b$. Several decades passed before Folkman and Sanders extended this statement from two integers $a, b$ to arbitrarily long sequences such that all finite sums of distinct elements are monochromatic. A real breakthrough was made in 1974 by Hindman, who showed, in the same setting, the existence of a unique infinite increasing sequence such that all finite sums of distinct elements from it are monochromatic. An year later Galvin and Glazer found an elegant proof that exploits the existence of idempotent ultrafilters.
In this talk we will give an historical introduction mentioning the previous results and we will explore the case of exponentiation, firstly investigated by Sisto (2011) and then by Sahasrabudhe (2018). The latter’s main result states that, for every finite coloring of $\mathbb N$ there exist arbitrarily long sequences such that all finite products and exponentiations of distinct elements are monochromatic. In our main theorem we prove that, for every finite coloring of $\mathbb N$ there exists an infinite increasing sequence such that all finite exponentiations of distinct elements are monochromatic. In other words, we realise for exponentiation the passage from finite to infinite made by Hindman for sum.
Our main tools are central sets, that are elements of a very special type of ultrafilter, not only idempotent but also minimal.
We finally extend the theorem to a larger class of binary non-associative operations which somehow behave in the same manner as exponentiation.
This is a joint work with Mauro Di Nasso.
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