New exact values of the maximum size of graphs free of topological complete subgraphs

The extremal number ex(n;TKp)ex(n;TKp) denotes the maximum number of edges of a graph of order n   containing no topological complete graph TKpTKp as a subgraph. In this paper we give the exact value of the extremal number ex(n;TKp)ex(n;TKp) for ⌈(7n+7)/12⌉⩽p<⌈(2n+1)/3⌉⌈(7n+7)/12⌉⩽p<⌈(2n+1)/3⌉ provided that n-p⩾15n-p⩾15. Furthermore, we find the corresponding extremal graphs for n-p⩾17n-p⩾17.