Large dense neighbourhoods and turán's theorem

Let t_r(n) denote the number of edges in the complete r-partite graph on n vertices whose colour classes are as equal in size as possible. It is proved that if G is a simple graph on n vertices and more than t_r(n) edges, and if v is any vertex of maximum degree m in G, then the subgraph of G induced by the neighbours of v has more than t_r?1(m) edges. Examples are given to demonstrate the sharpness of this theorem, and its algorithmic implications are noted.