• nonstandard analysis
• Partition regular
• ultrafilters

Ramsey Theory and partition regularity problems are interesting settings of combinatorics that investigate structural properties of families of sets. More precisely, a collection of a sets of A, namely F, is partition regular on the set A if, whenever A is finitely partitioned in C_1,...,C_r, then there exists an index j in {1,...,r} and an element of F contained in C_j.
Our interest is focused on diophantine equations. In particular we answer to the following question:
given a polynomial P, is P partition regular over the natural numbers?
This means: given a finite partition (or colouring) of natural numbers, can we find monochromatic solutions of P?

The thesis is structured in four chapters.
The first chapter lays the foundations of the rest of the thesis. It starts with the theory of ultrafilters which are important and multifaced mathematical objects, whose definition can be formulated in several languages: from set theory, as maximal families of closed under finite intersection sets, to measure theory, as {0,1}-valued finitely additive measures on a given space, to algebra as maximal ideals of ring of function F^I.
The chapter continues with a brief dissertation about nonstandard analysis that was created in the early 1960s by the mathematician Abraham Robinson. In particular we focus on hypernatural numbers and their properties to prove the main results of this thesis.
We show that the theory of ultrafilters and nonstandard analysis are strictly connected, and they have many applications in other fields of mathematics, as combinatorics or topology.

Though this, in the second chapter we can prove some important well-known results that concern partition regularity. We focus our attention on Ramsey Theory, a branch of combinatorics that studies the conditions under which order must appear. Typically, Ramsey problems are connected to questions of the form: how many elements of a given structure should there be to make true a particular property?
The begin of this theory is dated back to 1928 when Frank Plumpton Ramsey published his paper "On a problem of formal logic". The paper has led to a large area of combinatorics now known as Ramsey Theory and several important results arose from it in the last century. The most important results that are relevant to our purposes are: Schur's Theorem (1916), Rado's Theorem (1933) that gives a characterisation of the homogeneous systems to which a monochromatic solution can be found in any finite colouring of the natural numbers, Van der Waerden's Theorem (1927), Hindman’s Theorem (1974), and Milliken-Taylor's Theorem (1975).


Rado's Theorem completely settled the characterisation of partition regularity of the linear polynomials, and it is the starting point from which the heart of this thesis develops: the partition regularity of nonlinear equations. Actually, the third chapter is dedicated to proving the partition regularity of a few particular equations. Furthermore we give necessary conditions to say when a polynomial is partition regular. These conditions depend on Rado's Theorem and on the degree of the nonlinear variables.

In the last fourth chapter we investigate the non-partition regularity of large classes of nonlinear equations. Starting from two simple non partition regular polynomials, x^2+y^2-z and x+y-z^2, we aim at extending these examples. The first step toward the generalisation is to modify the exponents: we prove that the equations x^n+y^n=z^k and x^n+y^m=z^k with n,m,k mutually distinct are non-partition regular.
Subsequently we increase the numbers of variables and we prove that also the following equations are non-partition regular:
x_1^n+...+x_m^n = y^k with m>1 and k<n, or k>n and there exists a prime p that divides m and p^{k-n} does not divide m;
x_1^n+...+x_m^n = y^{n+1}.

In the end considering polynomials with coefficients c_j not equal to 1, under suitable conditions on the c_j, we have that the two following equations are non-partition regular:
c_1x_1^n+...+c_nx_m^n = y^k, with k<n, m>1;
c_1x_1^n+...+c_nx_m^n = y^{n+1}, with m>1.