Generalized Turán Problem for Directed Cycles

ABSTRACT For integers , let denote the maximum number of directed cycles of length in any oriented graph on vertices which does not contain a directed cycle of length . We establish the order of magnitude of for every and and determine its value up to a lower error term when and is large enough. Additionally, we calculate the value of for some other specific pairs showing that a diverse class of extremal constructions can appear for small values of .