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 $2^{O(\mathrm{La}^*(n,P))}$, where $\mathrm{La}^*(n,P)$ denotes the the largest size of an induced $P$-free subfamily of $2^{[n]}$. We also obtain related supersaturation results.