58.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 58

References:

erdosproblems.com/58
[Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.
[GaHuMa21] Gao, Jun and Huo, Qingyi and Ma, Jie, A strengthening on odd cycles in graphs of given chromatic number. SIAM J. Discrete Math. (2021), 2317-2327.
[Gy92] Gyárfás, A., Graphs with k odd cycle lengths. Discrete Math. (1992), 41-48.

Imports (1)

FormalConjecturesUtil

theorem Erdos58.erdos_58 If $G$ is a graph which contains odd cycles of $\leq k$ different lengths then $\chi(G)\leq 2k+2$, with equality if and only if $G$ contains $K_{2k+2}$ sorry line 42 source
Source documentation
If $G$ is a graph which contains odd cycles of $\leq k$ different lengths then $\chi(G)\leq 2k+2$, with equality if and only if $G$ contains $K_{2k+2}$.
Conjectured by Bollobás and Erdős. Proved by Gyárfás [Gy92].

depends on sorry what this means
Statement uses: SimpleGraph, Nat, Set.Finite, SimpleGraph.oddCycleLengths, LE.le, Set.ncard, And, ENat, SimpleGraph.chromaticNumber, HAdd.hAdd, HMul.hMul, OfNat.ofNat, Iff, Eq, SimpleGraph.IsContained, Fin, SimpleGraph.completeGraph
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_58 :
    ∀ {V : Type*} (G : SimpleGraph V) (k : ℕ),
      G.oddCycleLengths.Finite →
      G.oddCycleLengths.ncard ≤ k →
        G.chromaticNumber ≤ (2 * k + 2 : ℕ∞) ∧
        (G.chromaticNumber = (2 * k + 2 : ℕ∞) ↔
          (SimpleGraph.completeGraph (Fin (2 * k + 2))).IsContained G) := by
…
```