What is the greatest number of edges in a simple graph on $v$ vertices that contains no cycles of length smaller than $k$? What are extremal graphs in this case? This is an example of so-called degenerate extremal problem of Turan type.
In this talk I will
- present a short survey of old and new results on extremal problems of Turan type;
- discuss an algebraic method of constructing some best known examples of graphs with many edges and without certain small cycles;
- discuss some applications of the algebraic method to several problems of extremal graph theory (some of them have already been published but three are new).