Dense neighbourhoods and Turán's theorem

We prove the following extension of Turan's theorem, conjectured by Erdos. Let t r(n) be the number of edges in the r -partite Turan graph T r (n) of order n , and suppose G is a graph of order n with at least t r (n) edges. Then either G=T r (n) or else there is a vertex x such that the subgraph spanned by the neighbours of x contains at least t r−1 (d) +1 edges, where d is the degree of x . Furthermore d>c r n , where c r is a constant.