A finite list covers all ordered vertex pairs

Proved

WilliamsonShmoys.finite_vertex_pair_list__p2m_1f4c177d_eb20_4750_9a13_0c928226c3d6_a1

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For a finite type, there is a finite list containing every ordered pair of vertices.

Preamble

import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finite_vertex_pair_list__p2m_1f4c177d_eb20_4750_9a13_0c928226c3d6_a1
    {V : Type} [Fintype V] :
    ∃ pairs : List (V × V), ∀ u v : V, (u, v) ∈ pairs := by sorry

Source

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).

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