Independence and counting problems in Combinatorics and Number theory

The method ofhypergraph containers has become a very important tool for dealing with problems which can be phrased in the language of independent sets in hypergraphs. This method has applications to numerous problems in combinatorics and other areas. In this thesis we consider examples of such problems; in particular problems concerning sets avoiding solutions to a given system of linear equations L (known as L-free sets) or graphs avoiding copies of a given graph H (H-free graphs). First we attack a number of questions relating to L-free sets. For example, we give various bounds on the number of maximal L-free subsets of [n] for three-variable homogeneous linear equations L. We then use containers to prove results corresponding to problems concerning tuples of disjoint independentsets in hypergraphs. In particular we generalise the random Ramsey theorem of Rodl and Rucinski by providing a resilience analogue. We obtain similar results for L-free sets. Finally we consider the Maker-Breaker game where Maker's aim is to obtain a solution to a given system of linear equations L amongst a random set of integers. We determine the threshold probability for this game for a large class of systems L