Large joints in graphs

We show that if r ≥ s ≥ 2 , n > r 8 , and G is a graph of order n containing as many r -cliques as the r -partite Turán graph of order n , then G has more than n r − 1 / r 2 r + 12 cliques sharing a common edge unless G is isomorphic to the r -partite Turán graph of order n . This structural result generalizes a previous result that has been useful in extremal graph theory.