LTCC Course: Graph Theory October 2013 §5 Ramsey Theory and Regularity

Ramsey theory is about results in the following style: no matter how “chaotic” the overall structure is, if we look at the right (usually small) piece of the structure, we will find a pattern. The most standard version of this type of result is the graph Ramsey theorem. Here, the structure we examine is a colouring of the edges of a graph (not necessarily a proper colouring), and the pattern we are seeking is a set of k vertices such that the ( k 2 ) edges they span all have the same colour – this is a monochromatic copy of Kk. The theorem says that, whatever finite number c of colours we are provided with, if n is large enough, then in every colouring of the edges of Kn with c colours, there is a monochromatic copy of Kk. Theorem 2.1. If n ≥ 4k, then every 2-colouring of E(Kn) contains a monochromatic copy of Kk.