Forbidden induced subposets of given height

Let $P$ be a partially ordered set. The function $\mbox{La}^{\#}(n,P)$ denotes the size of the largest family $\mathcal{F}\subset 2^{[n]}$ that does not contain an induced copy of $P$. It was proved by Methuku and P\'alv\"olgyi that there exists a constant $C_{P}$ (depending only on $P$) such that $\mbox{La}^{\#}(n,P)<C_{P}\binom{n}{\lfloor n/2\rfloor}$. However, the order of the constant $C_{P}$ following from their proof is typically exponential in $|P|$. Here, we show that if the height of the poset is constant, this can be improved. We show that for every positive integer $h$ there exists a constant $c_{h}$ such that if $P$ has height at most $h$, then $$\mbox{La}^{\#}(n,P)\leq |P|^{c_{h}}\binom{n}{\lfloor n/2\rfloor}.$$ Our methods also immediately imply that similar bounds hold in grids as well. That is, we show that if $\mathcal{F}\subset [k]^{n}$ such that $\mathcal{F}$ does not contain an induced copy of $P$ and $n\geq 2|P|$, then $$|\mathcal{F}|\leq |P|^{c_{h}}w,$$ where $w$ is the width of $[k]^{n}$. A small part of our proof is to partition $2^{[n]}$ (or $[k]^{n}$) into certain fixed dimensional grids of large sides. We show that this special partition can be used to derive bounds in a number of other extremal set theoretical problems and their generalizations in grids, such as the size of families avoiding weak posets, Boolean algebras, or two distinct sets and their union. This might be of independent interest.