Largest digraphs contained in alln-tournaments

AbstractLetf(n) (resp.g(n)) be the largestm such that there is a digraph (resp. a spanning weakly connected digraph) onn-vertices andm edges which is a subgraph of every tournament onn-vertices. We prove that $$n\log _2 n - c_1 n \geqq f(n) \geqq g(n) \geqq n\log _2 n - c_2 n\log \log n.$$