Almost Given Length Cycles in Digraphs

A digraph is called k-cyclic if it cannot be made acyclic by removing less than k arcs. It is proved that for every ε > 0 there are constants K and δ so that for every d ∈ (0, δn), every ε n2-cyclic digraph with n vertices contains a directed cycle whose length is between d and d + K. A more general result of the same form is obtained for blow-ups of directed cycles.