TITLE: The maximal number of copies of a graph $T$ in an $H$-free graph
QUESTION [7 upvotes]: Problem. Let $T,H$ be fixed graphs with $H$ being a tree, not isomorphic to a subgraph of $T$. Let $ex(n,T,H)$ be the maximal number of copies of $T$ in an $H$-free graph on $n$ vertices. Is it always true that $ex(n,T,H)=\Theta(n^k)$ for $k=k(T,H)\in\mathbb N$?

(The problem was posed on 13.10.2016 by Clara Shikhelman. The promised prize for solution is "a bottle of wine in Tel-Aviv", see page 20 of Volume 1 of the Lviv Scottish Book).

REPLY [7 votes]: This was an answer to a previous version of the question, when $H$ was not claimed to be a tree.
It is well-known that a maximal number of edges in a $C_4$-free graph is $\Theta(n^{3/2})$ (since every pair of vertices is connected by at most one path of length 2, so the average square of a vertex degree is at most $\Theta(n)$; this is reached, e.g.,  by the incidence graph of a projective plane). Thus the conjecture fails for $H=C_4$, $T=K_2$.