The Forbidden Simplex: Maximal Edge Density of Planar Hasse Diagrams

We establish a tight upper bound on the number of edges in thecover graph of a finite partially ordered set whose Hasse diagramis planar.The transitive reduction that defines the Hasse diagram forcesevery 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 $n \geq 3$ verticessatisfies $|E| \leq 2n - 4$.We then show that this bound is tight: for every $n \geq 4$, thecomplete bipartite graph $K_{2, n-2}$ is realizable as a planarHasse diagram and achieves $|E| = 2n - 4$ exactly.Moreover, we prove that $K_{2, n-2}$ is the unique extremalstructure among planar Hasse diagrams of height~2, and that nosimplicial 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 diagramsfrom the simplicial meshes that arise in computational geometry,finite element methods, and other discrete geometric settings.