My main field of research is graph theory. While considering the occasional problem in other areas of graph theory and combinatorics, most of my research is concerned with two topics.
The first of these topics is graphs homomorphisms and constraint satisfaction problems. In this area one is faced with finite problems with finitely many local constraints, and must decide if there is a global solution satisfying all these local constraints. Our reserach focusses on the computational complexity of making this decision-- classifying the types of constraints under which we can do so in polynomial time. Such classifications require the use of graph theoretic, algebraic, and topological techniques.
The second is extremal graph theory, and in particular, Ramsey or Erdos-Ko-Rado type results. In these areas, we are faced with the classes graphs or hypergraphs satisfying various special properties, such as pairwise intersection, or non-avoidable substructures, and must determine the extremal (biggest, smallest, angriest, etc.) graphs in the class. Such problems require a mix of constructive and probabilistic techniques.