The Forbidden Simplex: Maximal Edge Density of Planar Hasse Diagrams

We establish a tight upper bound on the number of edges in the
cover graph of a finite partially ordered set whose Hasse diagram
is planar.
The transitive reduction that defines the Hasse diagram forces
every such graph to be triangle-free, yielding the clique bound
$\omega(H) \leq 2$.
Combining this constraint with Euler's formula for planar graphs,
we prove that every planar Hasse diagram on $n \geq 3$ vertices
satisfies $|E| \leq 2n - 4$.
We then show that this bound is tight: for every $n \geq 4$, the
complete bipartite graph $K_{2, n-2}$ is realizable as a planar
Hasse diagram and achieves $|E| = 2n - 4$ exactly.
Moreover, we prove that $K_{2, n-2}$ is the unique extremal
structure among planar Hasse diagrams of height~2, and that no
simplicial complex of dimension $\geq 2$ (triangle, tetrahedron,
or higher simplex) can appear as a subgraph of any Hasse diagram.
This ``forbidden simplex'' property distinguishes Hasse diagrams
from the simplicial meshes that arise in computational geometry,
finite element methods, and other discrete geometric settings.