Compare the planar edge bound with all possible pairs

Proved

WilliamsonShmoys.three_n_sub_six_le_choose_two__p2m_3be8fb19_1696_42e0_add6_8cf83fd59ba5_a2

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

approximation-algorithmsplanar-graphstreewidth

For every natural number n at least 3, the bound 3n−6 is at most the number of unordered pairs among n vertices.

Preamble

import Mathlib

Formal statement

theorem WilliamsonShmoys.three_n_sub_six_le_choose_two__p2m_3be8fb19_1696_42e0_add6_8cf83fd59ba5_a2
    (n : Nat) (hn : 3 ≤ n) :
    3 * n - 6 ≤ Nat.choose n 2 := 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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