Extremal problems for directed graphs

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