Large joints in graphs

We show that if r≥s≥2,n>r8r≥s≥2,n>r8, and GG is a graph of order nn containing as many rr-cliques as the rr-partite Turán graph of order nn, then GG has more than nr−1/r2r+12nr−1/r2r+12 cliques sharing a common edge unless GG is isomorphic to the rr-partite Turán graph of order nn. This structural result generalizes a previous result that has been useful in extremal graph theory.