Forbidden induced subposets in the grid

TL;DR

This paper generalizes a previous result by establishing an upper bound on the size of subsets in grid posets that avoid a specific induced subposet, linking it to the largest antichain size with a constant depending on the poset.

Contribution

It proves a generalized bound for induced subposet avoidance in grid posets, extending prior work to broader classes of posets and grid dimensions.

Findings

Bound on subset size proportional to largest antichain

Constant depends only on the poset

Applicable for any poset and grid dimension

Abstract

In this short paper, we prove the following generalization of a result of Methuku and P\'{a}lv\"{o}lgyi. Let $P$ be a poset, then there exists a constant $C_{P}$ with the following property. Let $k$ and $n$ be arbitrary positive integers such that $n$ is at least the dimension of $P$, and let $w$ be the size of the largest antichain of the grid $[k]^{n}$ endowed with the usual pointwise ordering. If $S$ is a subset of $[k]^{n}$ not containing an induced copy of $P$, then $|S|\leq C_{P}w$.