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The fractional-coloring counterexample

Lax342547.FractionalCounterexample · concepts/Lax342547/FractionalCounterexample.lean · lax-342547

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    Natural Language Statement

    Theorem

    Corollary 1.2 asserts that there are graphs of arbitrarily large order mm with independence number at most two and h(G)<26m/75+2/3<m/2≤χf(G)≤χ(G)h(G)<26m/75+2/3<m/2\le\chi_f(G)\le\chi(G). Here h(G)h(G) refers to ordinary clique minors.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    1import Lax342547.CliqueMinor
    2import Lax342547.FractionalColoring
    3
    4/-!
    5---
    6title: The fractional-coloring counterexample
    7type: theorem
    8---
    9Corollary 1.2 asserts that there are graphs of arbitrarily large order mm
    10with independence number at most two and
    11h(G)<26m/75+2/3<m/2≤χf(G)≤χ(G)h(G)<26m/75+2/3<m/2\le\chi_f(G)\le\chi(G).
    12Here h(G)h(G) refers to ordinary clique minors.
    13-/
    14
    15namespace Lax342547.FractionalCounterexample
    16
    17open CliqueMinor FractionalColoring
    18
    19axiom arbitrarily_large_fractional_counterexample (lowerBound : ℕ) :
    20 ∃ m : ℕ, lowerBound ≤ m ∧ 5 ≤ m ∧ ∃ G : SimpleGraph (Fin m),
    21 G.indepNum ≤ 2 ∧
    22 (hadwigerNumber G : ℝ) < 26 * (m : ℝ) / 75 + 2 / 3 ∧
    23 26 * (m : ℝ) / 75 + 2 / 3 < (m : ℝ) / 2 ∧
    24 (m : ℝ) / 2 ≤ fractionalChromaticNumber G ∧
    25 fractionalChromaticNumber G ≤ (G.chromaticNumber.toNat : ℝ)
    26
    27end Lax342547.FractionalCounterexample
    28
    Show Proof
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