Rosario MENNUNI (Università di Milano) will give a talk on
Ramsey Witnesses Part 2: Pairwise Problems
Abstract:
I will talk about recent joint work with M. Di Nasso, L. Luperi Baglini, M. Mamino and M. Ragosta, where we consider the following problem: given binary functions $f(x,y)$, $g(x,y)$ on the natural numbers, when is it the case that, for every finite colouring of the naturals, there is an increasing sequence $(a_i)$ such that all $f(a_i, a_j)$ and $g(a_i,a_j)$, for $i<j$, have the same colour?
We study this problem by introducing the notion of “Ramsey partition regularity”, and then characterising it in terms of certain ultrafilters related to tensor products and dubbed “Ramsey’s witnesses”, and in terms of nonstandard arithmetic. These characterisations are then used to solve various instances of the problem above, e.g. involving polynomials and exponentials.