596.lean`

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

9 declarations, 7 contain sorry, 0 declared as axioms

Erdős Problem 596

References:

erdosproblems.com/596
[Er87] Erdős, Some of my favourite problems in various branches of combinatorics, Mat. Lapok 1987.
[NeRo75] Nešetřil and Rödl, The Ramsey property for graphs with forbidden complete subgraphs, J. Combin. Theory B 20 (1976), 243--249.

Imports (1)

FormalConjecturesUtil

theorem Erdos596.erdos_596 Erdős Problem 596 (Erdős–Hajnal, [Er87]) sorry line 53 source
Source documentation
Erdős Problem 596 (Erdős–Hajnal, [Er87]). For which graph pairs $(G_1, G_2)$ is it true that
(1) for every $n \geq 1$ there is a graph $H$ without a $G_1$ such that any $n$-colouring of $H$'s edges contains a monochromatic $G_2$, and yet (2) for every graph $H$ without a $G_1$ there is an $\aleph_0$-colouring of $H$'s edges with no monochromatic $G_2$?
Erdős and Hajnal originally conjectured that no such pair exists; but $(C_4, C_6)$ witnesses it (Nešetřil–Rödl + Erdős–Hajnal). The full question is to characterise the class of all such pairs, recorded here as answer(sorry).
See Problem 595 for the specific case $(G_1, G_2) = (K_4, K_3)$.

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_596 :
    ∀ {U₁ U₂ : Type} (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂),
      IsErdosHajnalExceptional G₁ G₂ ↔
      (answer(sorry) : ∀ {U₁ U₂ : Type}, SimpleGraph U₁ → SimpleGraph U₂ → Prop) G₁ G₂ := by
…
```
theorem Erdos596.erdos_596.variants.exists_exceptional Erdős–Hajnal exceptional pairs exist — recorded as a known direction of erdos_596 sorry line 61 source
Source documentation
Erdős–Hajnal exceptional pairs exist — recorded as a known direction of erdos_596.

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_596.variants.exists_exceptional :
    ∃ (U₁ U₂ : Type) (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂),
      IsErdosHajnalExceptional G₁ G₂ := by
…
```
theorem Erdos596.erdos_596.variants.C4_free_countable_escape Every $C_4$-free graph is a countable union of trees (Erdős–Hajnal [Er87]); trees are acyclic, hence $C_6$-free, giving the countable Ramsey escape for $(C_4, C_6)$ sorry line 69 source
Source documentation
Every $C_4$-free graph is a countable union of trees (Erdős–Hajnal [Er87]); trees are acyclic, hence $C_6$-free, giving the countable Ramsey escape for $(C_4, C_6)$.

depends on sorry what this means
Statement uses: SimpleGraph.HasCountableRamseyEscape, Fin, OfNat.ofNat, Nat, SimpleGraph.cycleGraph
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_596.variants.C4_free_countable_escape :
    HasCountableRamseyEscape (cycleGraph 4) (cycleGraph 6) := by
…
```
theorem Erdos596.erdos_596.variants.C4_C6_finite_ramsey Nešetřil–Rödl [NeRo75]: for every $n \geq 1$ there is a $C_4$-free graph whose edges cannot be $n$-coloured without a monochromatic $C_6$ sorry line 76 source
Source documentation
Nešetřil–Rödl [NeRo75]: for every $n \geq 1$ there is a $C_4$-free graph whose edges cannot be $n$-coloured without a monochromatic $C_6$.

depends on sorry what this means
Statement uses: SimpleGraph.HasFiniteRamseyProperty, Fin, OfNat.ofNat, Nat, SimpleGraph.cycleGraph
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_596.variants.C4_C6_finite_ramsey :
    HasFiniteRamseyProperty (cycleGraph 4) (cycleGraph 6) := by
…
```
theorem Erdos596.erdos_596.variants.C4_C6_is_exceptional The pair $(C_4, C_6)$ is Erdős–Hajnal exceptional; combines C4_C6_finite_ramsey and C4_free_countable_escape line 83 source
Source documentation
The pair $(C_4, C_6)$ is Erdős–Hajnal exceptional; combines C4_C6_finite_ramsey and C4_free_countable_escape.

depends on sorry what this means
Statement uses: SimpleGraph.IsErdosHajnalExceptional, Fin, OfNat.ofNat, Nat, SimpleGraph.cycleGraph
Proof uses: And.intro, SimpleGraph.HasFiniteRamseyProperty, SimpleGraph.HasCountableRamseyEscape, Erdos596.erdos_596.variants.C4_C6_finite_ramsey, Erdos596.erdos_596.variants.C4_free_countable_escape
2 instances, coercions and auxiliary constants left out.
```
theorem erdos_596.variants.C4_C6_is_exceptional :
    IsErdosHajnalExceptional (cycleGraph 4) (cycleGraph 6) :=
…
```
theorem Erdos596.erdos_596.variants.original_conjecture_is_false The original Erdős–Hajnal conjecture (that no exceptional pair exists) is false — witnessed by $(C_4, C_6)$ via C4_C6_is_exceptional sorry line 90 source
Source documentation
The original Erdős–Hajnal conjecture (that no exceptional pair exists) is false — witnessed by $(C_4, C_6)$ via C4_C6_is_exceptional.

depends on sorry what this means
Statement uses: Iff, False, SimpleGraph, Not, SimpleGraph.IsErdosHajnalExceptional
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_596.variants.original_conjecture_is_false : answer(False) ↔
    ∀ {U₁ U₂ : Type} (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂),
      ¬IsErdosHajnalExceptional G₁ G₂ := by
…
```
theorem Erdos596.erdos_596.variants.K4_K3_exceptional_iff Whether $(K_4, K_3)$ is Erdős–Hajnal exceptional is precisely the content of Erdős Problem 595 sorry line 100 source
Source documentation
Whether $(K_4, K_3)$ is Erdős–Hajnal exceptional is precisely the content of Erdős Problem 595. The finite Ramsey property holds (Folkman 1970, Nešetřil–Rödl [NeRo75]); the open part is whether every $K_4$-free graph is a countable union of triangle-free graphs.

depends on sorry what this means
Statement uses: Iff, True, SimpleGraph.IsErdosHajnalExceptional, Fin, OfNat.ofNat, Nat, SimpleGraph.completeGraph
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_596.variants.K4_K3_exceptional_iff : answer(sorry) ↔
    IsErdosHajnalExceptional (completeGraph (Fin 4)) (completeGraph (Fin 3)) := by
…
```
theorem Erdos596.erdos_596.variants.K4_K3_finite_ramsey Folkman 1970 / Nešetřil–Rödl [NeRo75]: for every $n \geq 1$ there is a $K_4$-free graph whose edges cannot be $n$-coloured without a monochromatic triangle sorry line 107 source
Source documentation
Folkman 1970 / Nešetřil–Rödl [NeRo75]: for every $n \geq 1$ there is a $K_4$-free graph whose edges cannot be $n$-coloured without a monochromatic triangle.

depends on sorry what this means
Statement uses: SimpleGraph.HasFiniteRamseyProperty, Fin, OfNat.ofNat, Nat, SimpleGraph.completeGraph
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_596.variants.K4_K3_finite_ramsey :
    HasFiniteRamseyProperty (completeGraph (Fin 4)) (completeGraph (Fin 3)) := by
…
```
theorem Erdos596.erdos_596.test.empty_is_free The empty graph on Fin 0 is Free of any nontrivial subgraph (vacuous) line 116 source
Source documentation
The empty graph on Fin 0 is Free of any nontrivial subgraph (vacuous). This is the simplest non-trivial witness to G₁.Free H appearing in HasFiniteRamseyProperty.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph.Free, Fin, OfNat.ofNat, Nat, SimpleGraph.cycleGraph, Bot.bot, SimpleGraph
Proof uses: SimpleGraph.IsContained, False, SimpleGraph.Copy, Fin.elim0, Function.Embedding, SimpleGraph.Copy.toEmbedding
15 instances, coercions and auxiliary constants left out.
```
theorem erdos_596.test.empty_is_free :
    (cycleGraph 4).Free (⊥ : SimpleGraph (Fin 0)) :=
…
```