An Extremal Result for Subgraphs with Few Edges

We prove that whenever the edge number of a graph of order n ≥ 517 ensures that it contains every complete graph and every forest with at most n vertices and at most m edges, then the graph contains every graph with at most n vertices and m edges if m < n. The required edge numbers for complete graphs and for forests are derived from Turán′s Theorem and from an earlier result of the author, respectively.