Catalog/Sparsity/NDSubpolynomialWcol/Contract.lean
| 1 | import Catalog.Sparsity.Preliminaries.Contract |
| 2 | import Catalog.Sparsity.ColoringNumbers.Contract |
| 3 | import Catalog.Sparsity.NowhereDense.Contract |
| 4 | import Mathlib.Analysis.SpecialFunctions.Pow.Real |
| 5 | |
| 6 | open Catalog.Sparsity.ColoringNumbers |
| 7 | open Catalog.Sparsity.NowhereDense |
| 8 | open Catalog.Sparsity.Preliminaries |
| 9 | |
| 10 | namespace Catalog.Sparsity.NDSubpolynomialWcol |
| 11 | |
| 12 | /-- Theorem 3.4/ch2: Nowhere dense classes have subpolynomial weak coloring |
| 13 | number. |
| 14 | |
| 15 | For any ND class C, r ∈ ℕ, ε > 0, there exists a constant f such that |
| 16 | every G ∈ C admits an ordering with wcol_r(G,σ) ≤ f · |V(G)|^ε. -/ |
| 17 | axiom nd_subpolynomial_wcol (C : GraphClass) (hC : IsNowhereDense C) |
| 18 | (r : ℕ) (ε : ℝ) (hε : 0 < ε) : |
| 19 | ∃ f : ℝ, 0 < f ∧ ∀ {V : Type} [DecidableEq V] [Fintype V] |
| 20 | (G : SimpleGraph V), |
| 21 | C G → ∃ (ord : LinearOrder V), |
| 22 | letI := ord; (↑(wcol G r) : ℝ) ≤ f * (Fintype.card V : ℝ) ^ ε |
| 23 | |
| 24 | end Catalog.Sparsity.NDSubpolynomialWcol |
| 25 |