944.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

8 declarations, 7 contain sorry, 0 declared as axioms

Erdős Problem 944

Reference: erdosproblems.com/944

Imports (1)

FormalConjecturesUtil

def Erdos944.SimpleGraph.IsErdos944 The predicate that graph $G$ with chromatic number $k$ is such that every vertex is critical, yet every critical set of edges has size $>r$ line 39 source
Source documentation
The predicate that graph $G$ with chromatic number $k$ is such that every vertex is critical, yet every critical set of edges has size $>r$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Nat
Proof uses: And, SimpleGraph.IsCritical, Set, Sym2, SimpleGraph.IsCriticalEdges, LT.lt, Set.ncard
1 instances, coercions and auxiliary constants left out.
```
def SimpleGraph.IsErdos944 (G : SimpleGraph V) (k r : ℕ) : Prop :=  G.IsCritical k ∧
…
```
theorem Erdos944.erdos_944 Let $k \ge 4$ and $r\ge 1$ sorry line 47 source
Source documentation
Let $k \ge 4$ and $r\ge 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Exists, SimpleGraph, Erdos944.SimpleGraph.IsErdos944
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_944 :
    answer(sorry) ↔ ∀ k ≥ 4, ∀ r ≥ 1, ∃ (V : Type u) (G : SimpleGraph V), G.IsErdos944 k r := by
…
```
theorem Erdos944.erdos_944.variants.dirac_conjecture Let $k \ge 4$ sorry line 58 source
Source documentation
Let $k \ge 4$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>1$?
This was conjectured by Dirac in 1970.

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Exists, SimpleGraph, Erdos944.SimpleGraph.IsErdos944
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_944.variants.dirac_conjecture :
    answer(True) ↔ ∀ k ≥ 4, ∃ (V : Type u) (G : SimpleGraph V), G.IsErdos944 k 1 := by
…
```
theorem Erdos944.erdos_944.variants.dirac_conjecture.k_eq_5 Documented declaration: k_eq_5 sorry line 71 source
Source documentation
Dirac's conjecture was proved, for $k=5$: There exists a graph $G$ with chromatic number $5$, such that every vertex is critical, yet every critical set of edges has size $>1$, or in other words: has no critical edge.
[Br92] Brown, Jason I., A vertex critical graph without critical edges. Discrete Math. (1992), 99--101

depends on sorry what this means
Statement uses: Exists, SimpleGraph, Erdos944.SimpleGraph.IsErdos944, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_944.variants.dirac_conjecture.k_eq_5 :
    ∃ (V : Type u) (G : SimpleGraph V), G.IsErdos944 5 1 := by
…
```
theorem Erdos944.erdos_944.variants.dirac_conjecture.k_sub_one_not_prime Lattanzio [La02] proved there exist $k$-critical graphs without critical edges for all $k$ such that $k - 1$ is not prime sorry line 82 source
Source documentation
Lattanzio [La02] proved there exist $k$-critical graphs without critical edges for all $k$ such that $k - 1$ is not prime.
[La02] Lattanzio, John J., A note on a conjecture of {D}irac. Discrete Math. (2002), 323--330

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Not, Nat.Prime, HSub.hSub, Exists, SimpleGraph, Erdos944.SimpleGraph.IsErdos944
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_944.variants.dirac_conjecture.k_sub_one_not_prime (k : ℕ) (hk : 4 ≤ k)
    (h : ¬ (k - 1).Prime) : ∃ (V : Type u) (G : SimpleGraph V), G.IsErdos944 k 1 := by
…
```
theorem Erdos944.erdos_944.variants.dirac_conjecture.k_ge_five Jensen [Je02] gave an construction for $k$-critical graphs without any critical edges for all $k ≥ 5$ sorry line 92 source
Source documentation
Jensen [Je02] gave an construction for $k$-critical graphs without any critical edges for all $k ≥ 5$.
[Je02] Jensen, Tommy R., Dense critical and vertex-critical graphs. Discrete Math. (2002), 63--84.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, SimpleGraph, Erdos944.SimpleGraph.IsErdos944
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_944.variants.dirac_conjecture.k_ge_five (k : ℕ) (hk : 5 ≤ k) :
    ∃ (V : Type u) (G : SimpleGraph V), G.IsErdos944 k 1 := by
…
```
theorem Erdos944.erdos_944.variants.dirac_conjecture.k_eq_four sorry line 107 source
depends on sorry what this means
Statement uses: Iff, True, Exists, SimpleGraph, Erdos944.SimpleGraph.IsErdos944, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos944.erdos_944.variants.large_k_for_any_r Documented declaration: large_k_for_any_r sorry line 119 source
Source documentation
Martinsson and Steiner [MaSt25] proved for every $r \ge 1$ if $k$ is sufficiently large, depending on $r$, there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$.
[MaSt25] Martinsson, Anders and Steiner, Raphael, Vertex-critical graphs far from edge-criticality. Combin. Probab. Comput. (2025), 151--157

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Filter.Eventually, Exists, SimpleGraph, Erdos944.SimpleGraph.IsErdos944, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_944.variants.large_k_for_any_r (r : ℕ) (hr : 1 ≤ r) : ∀ᶠ k in Filter.atTop,
    ∃ (V : Type u) (G : SimpleGraph V), G.IsErdos944 k r := by
…
```