Forbidden hypermatrices imply general bounds on induced forbidden subposet problems

TL;DR

This paper establishes a universal bound on the size of subset families avoiding a given induced subposet, using forbidden hypermatrix techniques and a higher-dimensional Marcus-Tardos theorem variant.

Contribution

It proves a conjecture linking induced subposet avoidance to forbidden hypermatrix bounds, introducing a new proof of the higher-dimensional Marcus-Tardos theorem.

Findings

Bound C is universal for all posets P

Connection established between poset problems and forbidden hypermatrices

New proof provided for the higher-dimensional Marcus-Tardos theorem

Abstract

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.