A nonempty predicate on a finite partial order has a maximal member
ProvedWilliamsonShmoys.finite_predicate_has_maximal_extension__p2m_90a1d3cf_69db_4650_95dd_fb78166e4c20_a1approximation-algorithmsplanar-graphstreewidth
Every nonempty predicate on a finite partially ordered type contains an element that has no strictly greater member in that predicate.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.finite_predicate_has_maximal_extension__p2m_90a1d3cf_69db_4650_95dd_fb78166e4c20_a1
{α : Type} [PartialOrder α] [Finite α]
(S : α → Prop) (a : α) (ha : S a) :
∃ b, S b ∧ ∀ c, b ≤ c → S c → c = b := by sorrySource
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).