On the structure of linear graphs

Introduction. If the numbers of vertices and edges of a (linear) graph are suitably restricted, it is to be expected that something can be said about the configurations which the graph contains. As far as we know the first result in this direction is due to Turân. He proved that a graph with kn vertices and Ck,2n+1 edges always contains a complete graph of order k + 1. We shall here prove one such theorem (which arose originally out of a topological problem), and then list (without proofs) several other theorems and conjectures of this nature.