Maximal element of a nonempty finite subset of a partial order

Proved

WilliamsonShmoys.finite_finset_maximal__p2m_d87be54e_bfcb_4aba_9277_804ad42ab6ab_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 with no strictly larger element in that subset.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finite_finset_maximal__p2m_d87be54e_bfcb_4aba_9277_804ad42ab6ab_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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