Abstract

We shall consider graphs (hypergraphs) without loops and multiple edges. Let Ybe a family of so called prohibited graphs and ex (n, Y) denote the maximum number of edges (hyperedges) a graph (hypergraph) on n vertices can have without containing subgraphs from Y A graph (hypergraph) will be called supersaturated if it has more edges than ex (n, Y). If G has n vertices and ex (n, Y)=k edges (hyperedges), then it always contains prohibited subgraphs. The basic question investigated here is: At least how many copies of L E Y must occur in a graph G " on n vertices with ex (n, Y)+k edges (hy peredges)? Notation. In this paper we shall consider only graphs and hypergraphs without loops and multiple edges, and all hypergraphs will be uniform. If G is a graph or hypergraph, e(G), v(G) and y(G) will denote the number of edges, vertices and the chromatic number of G, respectively. The first upper index (without brackets) will denote the number of vertices: G", S", T ",P are graphs of order n. Kph)(m r,..., m p) denotes the h-uniform hypergraph with m,+...+mp vertices partitioned into classes Cl,..., C p, where JQ=mi (i=1,..., p) and the hyperedges of this graph are those h-tuples, which have at most one vertex in each C i. For h=2 KP (ni l,..., nt p) is the ordinary complete p-partite graph. In some of our assertions we shall say e.g. that "changing o(nl) edges in G "... ". (Of course, o(-) cannot be applied to one graph.) As a matter of fact, in such cases we always consider a sequence of graphs G " and n--.

Citations