A collection of problems in extremal combinatorics

Extremal combinatorics is concerned with how large or small a combinatorial structure can be if we insist it satisfies certain properties. In this thesis we investigate four different problems in extremal combinatorics, each with its own unique flavour. We begin by examining a graph saturation problem. We say a graph G is H-saturated if G contains no copy of H as a subgraph, but the addition of any new edge to G creates a copy of H. We look at how few edges a Kpsaturated graph can have when we place certain conditions on its minimum degree. We look at a problem in Ramsey Theory. The k-colour Ramsey number Rk(H) of a graph H is defined as the least integer n such that every kcolouring of Kn contains a monochromatic copy of H. For an integer r > 3 let Cr denote the cycle on r vertices. By studying a problem related to colourings without short odd cycles, we prove new lower bounds for Rk(Cr) when r is odd. Bootstrap percolation is a process in graphs that can be used to model how infection spreads through a community. We say a set of vertices in a graph percolates if, when this set of vertices start off as infected, the whole graph ends up infected. We study minimal percolating sets, that is, percolating sets with no proper percolating subsets. In particular, we investigate if there is any relation between the smallest and the largest minimal percolating sets in bounded degree graph sequences. A tournament is a complete graph where every edge has been given an orientation. We look at the maximum number of directed k-cycles a tournament can have and investigate when there exist tournaments with many more k-cycles than expected in a random tournament.