751.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

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 751

References:

erdosproblems.com/751
[Er94b] Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261--269.
[BoVi98] Bondy, J. A. and Vince, A., Cycles in a graph whose lengths differ by one or two. J. Graph Theory (1998), 11--15.

Imports (1)

FormalConjecturesUtil

theorem Erdos751.erdos_751.parts.i Let $G$ be a graph with chromatic number $\chi(G)=4$ sorry line 50 source
Source documentation
Let $G$ be a graph with chromatic number $\chi(G)=4$. If $m_1<m_2<\cdots$ are the lengths of the cycles in $G$ then can $\min(m_{i+1}-m_i)$ be arbitrarily large?
The answer is no: Bondy and Vince [BoVi98] proved that every graph with minimum degree at least $3$ has two cycles whose lengths differ by at most $2$, and hence the same is true for every graph with chromatic number $4$.
erdos_751.variants.finite below carries the Lean proof. It assumes a finite vertex type, where this quantifies over any V : Type, and the two are joined by de Bruijn-Erdős: a graph that is not $3$-colourable has a finite subgraph that is not $3$-colourable, and cycles of that subgraph are cycles of the whole. Mathlib does not have de Bruijn-Erdős, so that step is not formalised here.

depends on sorry what this means
Statement uses: Iff, False, Nat, Exists, SimpleGraph, And, Eq, ENat, SimpleGraph.chromaticNumber, OfNat.ofNat, Membership.mem, Set, SimpleGraph.cycleLengths, LT.lt, LE.le, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_751.parts.i :
    answer(False) ↔
      ∀ k : ℕ, ∃ (V : Type) (G : SimpleGraph V), G.chromaticNumber = 4 ∧
        ∀ m ∈ G.cycleLengths, ∀ m' ∈ G.cycleLengths, m < m' → m + k ≤ m' := by
…
```
theorem Erdos751.erdos_751.parts.ii Let $G$ be a graph with chromatic number $\chi(G)=4$ sorry line 66 source
Source documentation
Let $G$ be a graph with chromatic number $\chi(G)=4$. If $m_1<m_2<\cdots$ are the lengths of the cycles in $G$ then can $\min(m_{i+1}-m_i)$ be arbitrarily large? Can this happen if the girth of $G$ is large?
The answer is no: Bondy and Vince [BoVi98] proved that every graph with minimum degree at least $3$ has two cycles whose lengths differ by at most $2$, and hence the same is true for every graph with chromatic number $4$.

depends on sorry what this means
Statement uses: Iff, False, Nat, Exists, SimpleGraph, And, Eq, ENat, SimpleGraph.chromaticNumber, OfNat.ofNat, LE.le, SimpleGraph.girth, Membership.mem, Set, SimpleGraph.cycleLengths, LT.lt, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_751.parts.ii :
    answer(False) ↔
      ∀ k g : ℕ, ∃ (V : Type) (G : SimpleGraph V), G.chromaticNumber = 4 ∧ g ≤ G.girth ∧
        ∀ m ∈ G.cycleLengths, ∀ m' ∈ G.cycleLengths, m < m' → m + k ≤ m' := by
…
```
theorem Erdos751.erdos_751.variants.bondy_vince Bondy and Vince [BoVi98] proved that every graph with minimum degree at least $3$ has two cycles whose lengths differ by at most $2$ sorry line 77 source
Source documentation
Bondy and Vince [BoVi98] proved that every graph with minimum degree at least $3$ has two cycles whose lengths differ by at most $2$.

depends on sorry what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, LE.le, Nat, OfNat.ofNat, SimpleGraph.minDegree, Exists, And, Membership.mem, Set, SimpleGraph.cycleLengths, LT.lt, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_751.variants.bondy_vince {V : Type*} [Fintype V] (G : SimpleGraph V)
    [DecidableRel G.Adj] (hG : 3 ≤ G.minDegree) :
    ∃ m ∈ G.cycleLengths, ∃ m' ∈ G.cycleLengths, m < m' ∧ m' ≤ m + 2 := by
…
```
theorem Erdos751.erdos_751.variants.finite sorry line 95 source
depends on sorry what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, LE.le, ENat, OfNat.ofNat, Nat, SimpleGraph.chromaticNumber, Exists, And, Membership.mem, Set, SimpleGraph.cycleLengths, LT.lt, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.