  We prove that for every poset $P$, there is a constant $C$ such that the size
of any family of subsets of $[n]$ that does not contain $P$ as an induced
subposet is at most $C{\binom{n}{\lfloor\frac{n}{2}\rfloor}}$, settling a
conjecture of Katona, and Lu and Milans. We obtain this bound by establishing a
connection to the theory of forbidden submatrices and then applying a higher
dimensional variant of the Marcus-Tardos theorem, proved by Klazar and Marcus.
We also give a new proof of their result.
