Ramsey and Universality Properties of Random Graphs

This thesis investigates the interplay between two branches of discrete mathematics: Ramsey theory and random graphs. The origins of Ramsey theory can already be found in the work of Hilbert from the early 20th century. However, it was a simple but profound result of Frank P. Ramsey from 1930 which marked the beginning of the new field in mathematics. Phrased in graph theory terms, Ramsey’s theorem states that for every `-uniform hypergraph H and sufficiently large complete `-uniform hypergraph K n , no matter how one colours the edges of K n with two colours there will always exist a monochromatic copy of H, that is a copy of H with all edges having the same colour. For short, we say that such K n is Ramsey for H. Many variations of the Ramsey property have been studied since then. Generally speaking, we will use the term Ramsey-type property as an umbrella term for any such property involving colourings. Even though it might appear as a naive statement at first, Ramsey’s theorem gave rise to a whole new field with far-reaching results. Perhaps even more important, many techniques and ideas that are now considered to be standard tools in the arsenal of a mathematician have been developed in order to tackle questions in Ramsey theory. Thus it is a fruitful playground whose problems serve as benchmarks for ideas.