Maximal element in a finite predicate
ProvedWilliamsonShmoys.finite_predicate_maximal__p2m_93fb4221_4029_4870_92af_deed270da66d_a1approximation-algorithmsplanar-graphstreewidth
Every nonempty predicate on a finite partially ordered type contains an element with no strictly larger element satisfying the predicate.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.finite_predicate_maximal__p2m_93fb4221_4029_4870_92af_deed270da66d_a1 {α : Type} [Fintype α] [PartialOrder α] (P : α → Prop) [DecidablePred P] (hP : ∃ a, P a) : ∃ a, P a ∧ ∀ b, P b → a ≤ b → b = a := 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).