Forbidden Hypermatrices Imply General Bounds on Induced Forbidden Subposet Problems

Abstract

We prove that for every posetP, there is a constantCPsuch that the size of any family of subsets of {1, 2, . . .,n} that does not containPas an induced subposet is at most$$C_P{\binom{n}{\lfloor\gfrac{n}{2}\rfloor}},$$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.