Graphs without minor complete subgraphs

The extremal number ex(n;MKp) denotes the maximum number of edges of a graph of order n containing no complete graph Kp as a minor. In this paper we give the exact value of the extremal number ex(n;MKp) for ⌈(5n+9)/8⌉⩽p⩽⌊(2n-1)/3⌋ provided that n-p⩾24. Indeed we show that this number is the size of the Turán Graph T2p-n-1(n) and this graph is the only extremal graph.