• nonstandard
• ultrafilter
• colorings
• monochromatic
• Hindman

In this thesis we analyse some of the most recent results abount Hindman's conjecture.

The conjecture, born in the 80s and still open, states that given any finite coloring (i.e. partitioning) of the naturals there are two numbers a and b such that {a,b,a+b, ab} is monochromatic.
A major result has been obtained in 2017 by Moreira, who managed to prove that in any finite coloring of the naturals there is a monochromatic set of the form {a, a+b, ab}. Following the ideas contained in this work, in 2022 Bowen managed to settle the conjecture in case of just two colors, and Bowen and Sabok settled it over the rationals.
Followin a complitely different path, which may be interesting to investigate more, Alweiss in 2023 proved a stronger version of Bowen-Sabok's theorem.

The thesis can be divided in two sections: the first one is devoted to give an overall look of these results. In the the second one, using tools coming from logic (nonstandard mathematics) we give a new proof of (in fact, part of) Alweiss' theorem.