Extremal graph problems with symmetrical extremal graphs. Additional chromatic conditions

The main result of this paper is that for a special, but rather wide class of ''sample graphs'', the extremal graphs, i.e. the graphs of n vertices without subgraphs isomorphic to the sample graph and having maximum number of edges under this condition, have very simple and symmetric structure. This result remains valid even in the case when the condition ''the graph does not contain the sample graph'' is replaced by the condition ''the graph does not contain the sample graph and its chromatic number is greater than t, where t is a fixed integer''. The results of this paper have a lot of different applications, a few of which are listed in Section 3.