On a problem in extremal graph theory

The number T∗(n,k) is the least positive integer such that every graph with n = (2k+1) + t vertices (t ≥ 0) and at least T∗(n,k) edges contains k mutually vertex-disjoint complete subgraphs S1, S2,…, Sk where Si has i vertices, 1 ≤ i ≤ k. Obviously T∗(n, k) ≥ T(n, k), the Turan number of edges for a Kk. It is shown that if n ≥ 98k2 then equality holds and that there is ϵ > 0 such that for (2k+1) ≤ n ≤ (2k+1) + ϵk2 inequality holds. Further T∗(n, k) is evaluated when k > k0(t).