Dimension of Posets
The dimension of a poset P is the least number
t such that ther is an order
preserving embedding from
P to R^{t}. There is also an
alternative definition via linear extensions
of P.
Dimension theory is a mature discipline with interesting connections to
various other branches of mathematics. To mention one: Dilworth proved that the
dimension of a distributive lattice equals the
width of the generating poset of the lattice (The famous duality theorem known
as Dilworth's Theorem originally was formulated as a lemma in this
context). To our work the following directions are most relevant:

Extremal combinatorics.

Schnyder woods.
Recall Schnyder's Theorem: A graph is planar if and only if its
incidence poset is a poset of dimension at most three.
The proof makes use of Schnyder
woods. The method has inspired a lot of research.