Extremal problems are at the hart of graph theory, constantly attracting a lot of attention. Probably the earliest such result was obtain more than 100 years ago by Mantel who computed the maximum number of edges in a triangle free graph on n vertices. This was generalized by Turan for all complete graphs and became a starting point of Extremal Graph Theory. In this talk we survey several classical open problems and results in this area and present some interesting applications of Extremal Graph Theory to other areas of mathematics.
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