Tournaments as strong subcontractions Original Research Article

We determine good bounds for the maximum size of a digraph which has no strong subcontraction to a tournament Tp of order p. In particular, we shall show that for a transitive tournament, denoted TTp then given any p and ε > 0, there exists n0 such that for all n ⩾ n0, if digraph D has order n and at least (2n)(1 − 1/(p − 1) + ε) edges, then D ≻s TTp, where ≻s denotes strong subcontraction. This uses a Turán type of argument. We also get some exact results for strong subcontraction of complete digraphs.