On the number of families avoiding a subposet

In this paper we show that for any poset $P$ that is not an antichain, the number of induced $P$-free families in the Boolean lattice $2^{[n]}$ is at most $2^{O({\mathrm{L}\mathrm{a}}^{\ast }(n,P))}$, where ${\mathrm{L}\mathrm{a}}^{\ast }(n,P)$ denotes the the largest size of an induced $P$-free subfamily of $2^{[n]}$. We also obtain related supersaturation results.