Dimension and cut vertices: an application of Ramsey theory

TL;DR

This paper explores the relationship between the dimension of posets and the graph-theoretic properties of their cover graphs, establishing a tight upper bound involving blocks and employing advanced combinatorial tools.

Contribution

It proves a new upper bound on the dimension of posets based on subposet properties and constructs examples demonstrating the bound's optimality, using the Product Ramsey Theorem.

Findings

The dimension of a poset is at most d+2 under given conditions.

The bound is proven to be tight with constructed examples.

The proof involves advanced combinatorial methods, including the Product Ramsey Theorem.

Abstract

Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every $d\geq 1$, if $P$ is a poset and the dimension of a subposet $B$ of $P$ is at most $d$ whenever the cover graph of $B$ is a block of the cover graph of $P$, then the dimension of $P$ is at most $d+2$. We also construct examples which show that this inequality is best possible. We consider the proof of the upper bound to be fairly elegant and relatively compact. However, we know of no simple proof for the lower bound, and our argument requires a powerful tool known as the Product Ramsey Theorem. As a consequence, our constructions involve posets of enormous size.