I will introduce Serre's uniformity problem and try to convey part of its importance by showing how a solution to it may be useful when dealing with some diophantine equations. In the end, I will present some of my contribution to the study of this question.

Ramsey theory is a branch of combinatorics whose aim is to find some "order" inside chaos, more precisely one is usually concerned in finding large monochromatic substructures in any finite colouring of a ‘rich’ structure.

A right R-module M is called a Utumi Module (U-module) if, whenever A and B are isomorphic submodules of M with A ∩ B = 0, there exist two summands K and T of M such that A is an essential submodule of K, B is an essential submodule of T and K ⊕ T is a direct summand of M . The class of U -modules is a simultaneous and strict generalization of three fundamental classes of modules; namely the quasi-continuous, the square-free and the automorphism-invariant modules.