Subgraphs with a positive minimum semidegree in digraphs with large outdegree

We prove that every n -vertex directed graph G with the minimum outdegree δ + ( G ) = d contains a subgraph H satisfying min δ + ( H ) , δ - ( H ) ≥ d ( d + 1 ) 2 n . We also show that if d = o ( n ) then this bound is asymptotically best possible.