The maximum number of triangles in [formula omitted]-free graphs.

The generalized Turán number ex (n , K s , H) is the maximum number of complete graph K s in an H -free graph on n vertices. Let F k be the friendship graph consisting of k triangles. Erdős and Sós (1976) determined the value of ex (n , K 3 , F 2). Alon and Shikhelman (2016) proved that ex (n , K 3 , F k) ≤ (9 k − 15) (k + 1) n. In this paper, by using a method developed by Chung and Frankl in hypergraph theory, we determine the exact value of ex (n , K 3 , F k) and the extremal graph for any F k when n ≥ 4 k 3 . [ABSTRACT FROM AUTHOR]