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 (H) 2. Combining this constraint with Euler's formula for planar graphs, we prove that every planar Hasse diagram on n 3 verticessatisfies |E| 2n - 4. We then show that this bound is tight: for every n 4, thecomplete bipartite graph K₂, ₍-₂ is realizable as a planarHasse diagram and achieves |E| = 2n - 4 exactly. Moreover, we prove that K₂, ₍-₂ is the unique extremalstructure among planar Hasse diagrams of height~2, and that nosimplicial complex of dimension 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.
Juan Pablo Silva Alvarado (Mon,) studied this question.