Graphs in which each m-tuple of vertices is adjacent to the same number n of other vertices

Let G be a finite graph in which each m-tuple of mutually distinct vertices is adjacent to exactly n other vertices. If m ≥3 then G is isomorphic to the complete m+n graph. For completeness we state the friendship theorem of Erdös, Rényi, and Sós and a theorem of Bose and Shrikhande, both of which deal with the case m=2.