Finite downward-closed properties admit maximal extensions

Proved

finite_simpleGraph_has_maximal_extension__p2m_f3863ecb_d96f_4f01_8e13_caf8f4700d63_a2

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type, any graph satisfying a downward-closed property has an extension satisfying that property which is maximal among such extensions.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem finite_simpleGraph_has_maximal_extension__p2m_f3863ecb_d96f_4f01_8e13_caf8f4700d63_a2
    {V : Type} [Fintype V]
    (P : SimpleGraph V → Prop) (G : SimpleGraph V)
    (hG : P G)
    (hdown : ∀ ⦃A B : SimpleGraph V⦄, A ≤ B → P B → P A) :
    ∃ H : SimpleGraph V,
      G ≤ H ∧ P H ∧
      ∀ K : SimpleGraph V, H ≤ K → P K → K = H := 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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