A nonempty finite subset of a partial order has a maximal member

Proved

WilliamsonShmoys.exists_maximal_mem_finset__p2m_5361ebef_f1bf_4ddf_9929_f0777c6d31bd_a1

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

approximation-algorithmsplanar-graphstreewidth

Every nonempty finite subset of a partial order contains an element that has no strictly larger member in that subset. Applied to the subset of graphs extending G and satisfying P, this gives the graph required by the target.

Preamble

import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_mem_finset__p2m_5361ebef_f1bf_4ddf_9929_f0777c6d31bd_a1 {α : Type} [Fintype α] [PartialOrder α] (S : Finset α) (hS : S.Nonempty) : ∃ a ∈ S, ∀ b ∈ S, a ≤ b → b ≤ a := 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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