Extremal graph theory is a branch of discrete mathematics and also
the central theme of extremal combinatorics. It studies graphs which are
extremal with respect to some parameter under certain restrictions.
A typical result in extremal graph theory is Mantel's theorem. It states
that the complete bipartite graph with equitable parts is the
graph the maximizes the number of edges among all triangle-free graphs.
One can say that extremal graph theory studies how local properties of
a graph influence its global structure.
Another fundamental topic in the field of combinatorics is the
probabilistic method, which is a nonconstructive method pioneered by
Paul Erdos for proving the existence of a prescribed kind of
mathematical object. One particular application of the probabilistic
method lies in the field of positional games, more specifically
Maker-Breaker games.
My dissertation focus mainly on various Turan-type questions and their
applications to other related areas as well as the employment of the
probabilistic method to study extremal problems and positional games.