A finite subset of a preorder has a maximal element

Proved

WilliamsonShmoys.finite_preorder_has_maximal__p2m_7d28393b_3717_42d0_a7e1_00f1426861ad_a2

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

approximation-algorithmsplanar-graphstreewidth

Every nonempty finite subset of a preorder has an element such that any element of the subset above it is also below it.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.finite_preorder_has_maximal__p2m_7d28393b_3717_42d0_a7e1_00f1426861ad_a2 {α : Type} [Fintype α] [Preorder α] (s : Finset α) (hne : s.Nonempty) : ∃ a : α, a ∈ s ∧ ∀ b : α, 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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