Topological minors of cover graphs and dimension

TL;DR

This paper proves that posets with bounded height and cover graphs excluding a fixed topological minor have bounded dimension, providing a combinatorial proof and explicit bounds, and supports a related conjecture about $(k+k)$-free posets.

Contribution

The paper offers a combinatorial proof for bounded dimension of certain posets, avoiding structural theorems, and introduces tools to support a conjecture on $(k+k)$-free posets.

Findings

Posets of bounded height with cover graphs excluding a fixed topological minor have bounded dimension.

A direct combinatorial method finds large clique subdivisions in cover graphs of high-dimension posets.

$(k+k)$-free posets with excluded topological minors contain only standard examples of bounded size.

Abstract

We show that posets of bounded height whose cover graphs exclude a fixed graph as a topological minor have bounded dimension. This result was already proven by Walczak. However, our argument is entirely combinatorial and does not rely on structural decomposition theorems. Given a poset with large dimension but bounded height, we directly find a large clique subdivision in its cover graph. Therefore, our proof is accessible to readers not familiar with topological graph theory, and it allows us to provide explicit upper bounds on the dimension. With the introduced tools we show a second result that is supporting a conjectured generalization of the previous result. We prove that $(k+k)$-free posets whose cover graphs exclude a fixed graph as a topological minor contain only standard examples of size bounded in terms of $k$.