... specifying conditions on the cardinality of the edge-set of the digraph that guarantee the existence of a directed simplecycle of length at most four. As usual in Tur´an type problems in directed ... done. the electronic journal of combinatorics 9 (2001), #N6 2giving inequality (a).To prove inequality (b), let Mbe the submatrix of M consisting of rows βm+1, ,mand all columns of M. Since ... ,2ρn.DenotethisblockM2.DenotebyM3 the block consisting of the rows i = |A+|+1, ,m and the columnsj =1, ,ρn.DenotebyM4 the block consisting of the rows i = |A+|+1, ,mand the columns j =2ρn+1, ,n.DenotebyM5 the block...