Forbidden induced subposets of given height

TL;DR

This paper improves bounds on the size of families avoiding induced subposets of fixed height in Boolean lattices and grids, showing polynomial dependence on the size of the poset for fixed height.

Contribution

It establishes that for posets of fixed height, the maximum size of families avoiding induced copies grows polynomially with the poset size, refining previous exponential bounds.

Findings

Bounded the size of families avoiding induced posets of fixed height by a polynomial in the poset size.

Extended bounds to grid structures, including families avoiding weak posets and Boolean algebras.

Introduced a partitioning method of the Boolean lattice into fixed-dimensional grids for extremal set problems.

Abstract

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.