UCLA

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.