In the original version of this paper Theorem 1.1 was mistakenly stated for all p. This was also observed by Pikhurko [Pi01] and by Schelp. The theorem is valid only in the linear, quadratic and cubic cases. Namely:

Let k > 2 be a positive integer, and let p=1,2,3. Then t_p(n,K_k)=e_p(T(n,k)), where T(n,k) is the Turán Graph.

Theorem 1.1 is sharp in the sense that for p \geq 4, t_p(n,K_k) is not obtained by the Turán graph. This can already be seen by the fact that the complete bipartite graph G=K_{\lfloor n/2-1 \rfloor, \lceil n/2+1 \rceil} has e_4(G) > e_4(T(n,3)).

A revised version of this paper addressing this comment can be found in [CaYu04].

References

[CaYu04] Y. Caro and R. Yuster, A Turán Type Problem Concerning the Powers of the Degrees of a Graph (revised) arXiv:math.CO/0401398 (2004).
[Pi01] O. Pikhurko, Remarks on a paper of Y. Caro and R. Yuster on a Turán problem arXiv:math.CO/0101235 (2001).