Forbidden hypermatrices imply general bounds on induced forbidden   subposet problems
  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. 
Abhishek Methuku and D\"om\"ot\"or P\'alv\"olgyi
math.CO