Simonovits Miklós: How to solve a Turán type extremal graph problem?

The main purpose of this lecture is to show that if one picks a
forbidden graph L ``at random'', then very often some old asymptotic
or quasi-asymptotic results provide almost immediate solution
of the corresponding Turán type extremal problem.

Among others, I shall "determine" the Turán extremal number of the
Petersen graph. We shall prove, e.g., that for n>n0 the (only)
extremal graph for the Petersen graph Q10 is the graph
Hn,3,3 obtained as follows. One fixes a Turán graph T(n−2),2
on n−2 vertices and joins two further vertices x and y to each
other and to all the vertices of T(n−2),2.

The solutions of the problems discussed in the lecture will lead to
some general theorems.

One case discussed by me is an old theorem, back from 1974, on extremal problems where the decomposition class contains a path.