• combinatorics
• mathematical logic
• nonstandard analysis
• Ramsey theory
• topological dynamics
• ultrafilters
• van der Waerden's theorem

Given a finite colouring of the natural numbers, a famous open problem in Ramsey Theory concerns the existence of a monochromatic set in the form {x, y, x + y, xy}, where x, y are distinct naturals. The recent article “Monochromatic sums and products in N” by Joel Moreira, makes an important advance by exhibiting numerous monochromatic families in very general forms, including {x, x+y, xy}. This result is achieved by combining the careful use of the polynomial van der Waerden Theorem and innovative technical constructions. The crucial tool is applying classical topological dynamics concepts to natural numbers. This thesis aims to retrace this proof and the polynomial van der Waerden theorem, investigating possible alternative routes and partial improvements of the results. In the first section the suitable topological space for solving the problem is studied. While Moreira’s article uses the compact space of ultrafilters over N, in this thesis we show how similar results follow by choosing the nonstandard space ∗N of hypernatural numbers. The non-standard setting allows for a simpler formalization of some basic notions of topological dynamics used in the proof, and the arguments appear more straightforward. In the second section, we provide a full proof of Moreira’s theorem, preceded by a simplified version that emphasizes the main ideas. In the third section, we retrace the polynomial van der Waerden theorem, whose proof makes use of PET (Polynomial Exhaustion Technique) induction and the van der Corput trick, and we study the possible extension of the result by adding constraints on the solutions. In the fourth section, we question the abundance of solutions to the van der Waerden problem, studying the extensibility of Mathias Beiglb¨ock’s result in “Arithmetic Progressions in abundance by combinatorial tools” to the case of polynomial families.