On a problem in extremal graph theory

The number T*(n, k) is the least positive integer such that every graph with n = (*:I) + t vertices (t > 0) and at least T*(n, k) edges contains k mutually vertex-disjoint complete subgraphs S, , S, ,..., Sk where S, has ivertices, I < i Q k. Obviously T*(n, k) > T(n, k), the Turan number of edges for a Ki . It is shown that if n > gkk” then equality holds and that there is c :0 such that for (“:I) < n G (‘:I) + Eke inequality holds. Further T*(n, k) is evaluated when k ’ k,(t).