Extremal Problems for Directed Graphs

We consider directed graphs without loops and multiple edges, where the exclusion of multiple edges means that two vertices cannot be joined by two edges of the same orientation. Let L I ,. . ., LQ be given digraphs. What is the maximum number of edges a digraph can have if it does not contain any L i as a subgraph and has given number of vertices? We shall prove the existence of a sequence of asymptotical extremal graphs having fairly simple structure. More exactly : There exist a matrix A = (ai,5)ij * and a sequence {S"} of graphs such that (i) the vertices of S" can be divided into classes Ct ,. . ., C, so that, if i =A j, each vertex of Ci is joined to each vertex of C ; by an edge oriented from C i to C; if and only if a,, ; = 2 ; the vertices of Ci are independent if ai ,, = 0 ; and otherwise ai , , = 1 and the digraph determined by C, is a complete acyclic digraph ; (ii) S" contains no L, but any graph having [en 2 ] more edges than S" must contain at least one Li. (Here the word graph is an "abbreviation" for "directed graph or digraph.") NOTATION The digraphs (= directed graphs) considered in this paper have neither loops nor multiple edges : a vertex cannot be joined to itself and the digraph cannot have two edges joining the vertices x and y and oriented