Planar posets, dimension, breadth and the number of minimal elements (Q304190)

An ordered set is called planar iff its diagram can be drawn in the plane without any edges crossing. The authors prove that a planar ordered set with \(t\) minimal elements has dimension \(\leq 2t+1\). They also show that, for every \(t\geq 3\), there is a planar ordered set with \(t\) minimal elements and dimension \(\geq t+3\). The lower bound construction can be modified to improve the state of knowledge on the conjecture that, for every \(d\geq 2\), there is an \(f(d)\) so that a planar ordered set of dimension \(f(d)\) must contain a standard example of dimension \(d\). Previously, it was only known that \(f\), if it exists, satisfies \(f(d)\geq d+2\). The authors give examples which show that, if \(f\) exists, then \(\lim _{d\to \infty } f(d)/d \geq 2\). After a thorough background discussion, proofs and examples are elementary and very accessible.