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 cornerstone theorem in the area due to Frank Ramsey states that for every positive integers k and m, whenever the k-subsets of the natural numbers are coloured with $m$ colours there always exists an infinite subset A of N such that A^(k), the set consisting of the k-subsets of A, is monochromatic. In 1975, Erdős, Simonovits and Sős, started a new line of research, commonly known as Anti-Ramsey theory. The problems in this area lie at the opposite end of Ramsey theory, in here one is interested in finding large totally multicoloured (rainbow) substructures.

    In this talk, we are interested in understanding what happens in between these two extremes. Given an edge colouring of a graph with a set of m colours, we say that the graph is m-coloured if each of the m colours is used. For an m-colouring Δ of N^(2), the complete graph on N, we denote by F_Δ the set all values γ for which there exists an infinite subset X \subset N such that X^(2) is γ-coloured. Properties of this set were first studied by Erickson in 1994. Here, we are interested in estimating the minimum size of F_Δ over all m-colourings Δ of N^(2). Indeed, we shall prove the following result. There exists an absolute constant α > 0$ such that for any positive integer m \neq ( {n \choose 2} +1,  {n \choose 2}+2: n > 2\right} , |F_{Δ| > (1+α)√(2m), for any m-colouring Δ of N^(2), thus proving a conjecture of Narayanan. This result is tight up to the order of the constant α.