Forbidden subposet problems in the grid

TL;DR

This paper investigates extremal and saturation problems for subposets within grid structures, exploring their connections to Boolean posets and forbidden submatrix problems, expanding the understanding of poset-free configurations.

Contribution

It introduces new results on forbidden subposet problems in grid posets, linking these to classical Boolean poset problems and submatrix avoidance.

Findings

Established bounds for subposet extremal functions in grid posets

Connected grid poset problems to Boolean poset and submatrix problems

Provided new insights into forbidden subposet configurations

Abstract

For posets $P$ and $Q$, extremal and saturation problems about weak and strong $P$-free subposets of $Q$ have been studied mostly in the case $Q$ is the Boolean poset $Q_n$, the poset of all subsets of an $n$-element set ordered by inclusion. In this paper, we study some instances of the problem with $Q$ being the grid, and its connections to the Boolean case and to the forbidden submatrix problem.