Extremal problems for directed graphs

AbstractWe 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 L1,…,La be given digraphs.What is the maximum number of edges a digraph can have if it does not contain and Li 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.j)i.j≤r and a sequence {Sn} of graphs such that 1.(i) the vertices of Sn can be divided into classes C1,…,Cr so that, if i ≠ j, each vertex of Ci is joined to each vertex of Cj by an edge oriented from Ci to Cj if and only if ai.j = 2; the vertices of Ci are independent if ai.i = 0; and otherwise ai.i = 1 and the digraph determined by Ci is a complete acyclic digraph;2.(ii) Sn contains no Li but any graph having [∈n2] more edges than Sn must contain at least one Li.(Here the word graph is an “abbreviation” for “directed graph or digraph.”