The generalized Turán problem of two intersecting cliques

For s < r, let B r,s be the graph consisting of two copies of K r , which share exactly s vertices.Denote by ex(n, K r , B r,s ) the maximum number of copies of K r in a B r,s -free graph on n vertices.About fifty years ago, Erdős and Sós determined ex(n, K 3 , B 3,1 ).Recently, Gowers and Janzer showed that ex(n, K r , B r,r-1 ) = n r-1-o(1) .It is a natural question to ask for ex(n, K r , B r,s ) for general r and s.In this paper, we mainly consider the problem for s = 1.Utilizing Zykov's symmetrization, we determine the exact value of ex(n, K 4 , B 4,1 ) for n ≥ 4. For r ≥ 5 and n sufficiently large, by the Füredi's structure theorem we show that ex(n, K r , B r,1 ) = N (K r-2 , T r-2 (n -2)), where N (K r-2 , T r-2 (n -2)) represents the number of copies of K r-2 in the (r -2)-partite Turán graph on n -2 vertices.