Counting copies of a fixed subgraph in F-free graphs

Abstract Fix graphs F and H and let ex ( n , H , F ) denote the maximum possible number of copies of the graph H in an n -vertex F -free graph. The systematic study of this function was initiated by Alon and Shikhelman [J. Comb. Theory, B. 121 (2016)]. In this paper, we give new general bounds concerning this generalized Turan function. We also determine ex ( n , P k , K 2 , t ) (where P k is a path on k vertices) and ex ( n , C k , K 2 , t ) asymptotically for every k and t . For example, it is shown that for t ≥ 2 and k ≥ 5 we have ex ( n , C k , K 2 , t ) = 1 2 k + o ( 1 ) ( t − 1 ) k ∕ 2 n k ∕ 2 . We also characterize the graphs F that cause the function ex ( n , C k , F ) to be linear in n . In the final section we discuss a connection between the function ex ( n , H , F ) and Berge hypergraph problems.

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