Tournaments as strong subcontractions

We determine good bounds for the maximum size of a digraph which has no strong subcontraction to a tournament T-p of order p. In particular, we shall show that for a transitive tournament, denoted TTp, then given any p and epsilon > 0, there exists no such that for all n greater than or equal to n(0), if digraph D has order n and at least (n/2)(1 - 1/(p - 1) + epsilon) edges, then D >(s) TTp, where >(s) denotes strong subcontraction. This uses a Turan type of argument. We also get some exact results for strong subcontraction of complete digraphs.