Description:

Title:
Arithmetic Ramsey Theory and nonstandard methods.

Abstract:
Arithmetic Ramsey theory is an area of combinatorics that focuses on the existence,
for any finite coloring (partition) of the natural numbers, of “monochromatic patterns”
defined by arithmetic operations.
A classical result is van der Waerden's Theorem, which states that monochromatic
arithmetic progressions of any prescribed (finite) length can always be found.
About infinite configurations, a fundamental result is Hindman's Theorem, which states
that one can always find an infinite sequence such that all sums of its distinct elements
are monochromatic.
Various techniques have been successfully applied to the problems of arithmetic Ramsey
Theory, including ergodic theory, topological dynamics, and algebra in the space of ultrafilters.
In this seminar, I will show some examples of applications in this field of 
nonstandard methods combined with the use of ultrafilters.
In particular, I will show a proof of van der Waerden's Theorem based on the technique 
of iterated non-standard extensions, and a proof of an extension of Hindman's Theorem 
that includes infinite monochromatic patterns of the form a, b, a+b, b/a.