Finite graph edge rank is bounded by the number of vertex pairs
ProvedWilliamsonShmoys.finite_graph_edge_rank_bound__p2m_9fc762a8_5f43_4315_941e_21ea4c59e3e4_a2approximation-algorithmsplanar-graphstreewidth
For a finite vertex type, the number of edges in a simple graph is at most the number of unordered vertex pairs.
Preamble
import Mathlib
Formal statement
theorem WilliamsonShmoys.finite_graph_edge_rank_bound__p2m_9fc762a8_5f43_4315_941e_21ea4c59e3e4_a2
{V : Type} [Fintype V] (G : SimpleGraph V) :
G.edgeSet.ncard ≤ Fintype.card (Sym2 V) := by sorrySource
David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).