918.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

9 declarations, 9 contain sorry, 0 declared as axioms

Erdős Problem 918

References:

erdosproblems.com/918
[ErHa68b] Erdős, P. and Hajnal, A., On chromatic number of infinite graphs. (1968), 83--98.
[Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.

Imports (1)

FormalConjecturesUtil

theorem Erdos918.erdos_918.parts.i Is there a graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on $\aleph_1$ vertices has chromatic number $\leq\aleph_0$? sorry line 41 source
Source documentation
Is there a graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on $\aleph_1$ vertices has chromatic number $\leq\aleph_0$?

depends on sorry what this means
Statement uses: Iff, True, Exists, SimpleGraph, And, Eq, Cardinal, Cardinal.mk, OrderEmbedding, Ordinal, Cardinal.aleph, OfNat.ofNat, Nat, SimpleGraph.chromaticCardinal, Set, LE.le, SimpleGraph.induce, Cardinal.aleph0
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_918.parts.i :
    answer(sorry) ↔ ∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ 2 ∧ G.chromaticCardinal = ℵ_ 2 ∧
      ∀ (W : Set V) (_ : #W = ℵ₁), (G.induce W).chromaticCardinal ≤ ℵ₀ := by
…
```
theorem Erdos918.erdos_918.parts.ii Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$? sorry line 52 source
Source documentation
Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$?

depends on sorry what this means
Statement uses: Iff, True, Exists, SimpleGraph, And, Eq, Cardinal, Cardinal.mk, OrderEmbedding, Ordinal, Cardinal.aleph, HAdd.hAdd, Ordinal.omega0, OfNat.ofNat, SimpleGraph.chromaticCardinal, Set, LE.le, SimpleGraph.induce, Cardinal.aleph0
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_918.parts.ii :
    answer(sorry) ↔
    ∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ (ω + 1) ∧ G.chromaticCardinal = ℵ₁ ∧
      ∀ (W : Set V) (_ : #W = ℵ_ ω), (G.induce W).chromaticCardinal ≤ ℵ₀ := by
…
```
theorem Erdos918.erdos_918.variants.all_subgraphs.parts.i Is there a graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on $\aleph_1$ vertices has chromatic number $\leq\aleph_0$? sorry line 67 source
Source documentation
Is there a graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on $\aleph_1$ vertices has chromatic number $\leq\aleph_0$?

depends on sorry what this means
Statement uses: Iff, True, Exists, SimpleGraph, And, Eq, Cardinal, Cardinal.mk, OrderEmbedding, Ordinal, Cardinal.aleph, OfNat.ofNat, Nat, SimpleGraph.chromaticCardinal, SimpleGraph.Subgraph, SimpleGraph.Subgraph.verts, LE.le, SimpleGraph.Subgraph.coe, Cardinal.aleph0
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_918.variants.all_subgraphs.parts.i :
    answer(sorry) ↔ ∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ 2 ∧ G.chromaticCardinal = ℵ_ 2 ∧
      ∀ (H : G.Subgraph) (_ : #H.verts = ℵ₁), H.coe.chromaticCardinal ≤ ℵ₀ := by
…
```
theorem Erdos918.erdos_918.variants.all_subgraphs.parts.ii Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$? sorry line 75 source
Source documentation
Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$?

depends on sorry what this means
Statement uses: Iff, True, Exists, SimpleGraph, And, Eq, Cardinal, Cardinal.mk, OrderEmbedding, Ordinal, Cardinal.aleph, HAdd.hAdd, Ordinal.omega0, OfNat.ofNat, SimpleGraph.chromaticCardinal, SimpleGraph.Subgraph, SimpleGraph.Subgraph.verts, LE.le, SimpleGraph.Subgraph.coe, Cardinal.aleph0
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_918.variants.all_subgraphs.parts.ii :
    answer(sorry) ↔
      ∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ (ω + 1) ∧ G.chromaticCardinal = ℵ₁ ∧
      ∀ (H : G.Subgraph) (_ : #H.verts = ℵ_ ω), H.coe.chromaticCardinal ≤ ℵ₀ := by
…
```
theorem Erdos918.erdos_918.variants.erdos_hajnal A question of Erd\H{o}s and Hajnal [ErHa68b], who proved, assuming the generalized continuum hypothesis, that for every finite sorry line 89 source
Source documentation
A question of Erd\H{o}s and Hajnal [ErHa68b], who proved, assuming the generalized continuum hypothesis, that for every finite $k \geq 1$ there is a graph with chromatic number $\aleph_1$ and $\aleph_k$ vertices where each subgraph on less than $\aleph_k$ vertices has chromatic number $\leq \aleph_0$.

depends on sorry what this means
Statement uses: Cardinal, LE.le, Cardinal.aleph0, Eq, HPow.hPow, OfNat.ofNat, Nat, Order.succ, LT.lt, Exists, SimpleGraph, And, Cardinal.mk, OrderEmbedding, Ordinal, Cardinal.aleph, SimpleGraph.chromaticCardinal, Set, SimpleGraph.induce
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_918.variants.erdos_hajnal (hGCH : ∀ c : Cardinal.{u}, ℵ₀ ≤ c → 2 ^ c = Order.succ c)
    (k : ℕ) (hk : 0 < k) : ∃ (V : Type u) (G : SimpleGraph V),
    #V = ℵ_ k ∧ G.chromaticCardinal = ℵ₁ ∧
      ∀ (W : Set V) (_ : #W < ℵ_ k), (G.induce W).chromaticCardinal ≤ ℵ₀ := by
…
```
Mentions: c
theorem Erdos918.erdos_918.variants.eq_aleph_0.parts.i In [Er69b] the questions are stated with $= \aleph_0$ rather than $\leq\aleph_0$ sorry line 105 source
Source documentation
In [Er69b] the questions are stated with $= \aleph_0$ rather than $\leq\aleph_0$. This is a likely typo since it can be shown that no such graph exists in this case.
This is the first question with induced subgraphs.

depends on sorry what this means
Statement uses: Not, Exists, SimpleGraph, And, Eq, Cardinal, Cardinal.mk, OrderEmbedding, Ordinal, Cardinal.aleph, OfNat.ofNat, Nat, SimpleGraph.chromaticCardinal, Set, SimpleGraph.induce, Cardinal.aleph0
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_918.variants.eq_aleph_0.parts.i :
    ¬∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ 2 ∧ G.chromaticCardinal = ℵ_ 2 ∧
      ∀ (W : Set V) (_ : #W = ℵ₁), (G.induce W).chromaticCardinal = ℵ₀ := by
…
```
theorem Erdos918.erdos_918.variants.eq_aleph_0_all_subgraphs.parts.i In [Er69b] the questions are stated with $= \aleph_0$ rather than $\leq\aleph_0$ sorry line 115 source
Source documentation
In [Er69b] the questions are stated with $= \aleph_0$ rather than $\leq\aleph_0$. This is a likely typo since it can be shown that no such graph exists in this case.
This is the first question with all subgraphs.

depends on sorry what this means
Statement uses: Not, Exists, SimpleGraph, And, Eq, Cardinal, Cardinal.mk, OrderEmbedding, Ordinal, Cardinal.aleph, OfNat.ofNat, Nat, SimpleGraph.chromaticCardinal, SimpleGraph.Subgraph, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, Cardinal.aleph0
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_918.variants.eq_aleph_0_all_subgraphs.parts.i :
    ¬∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ 2 ∧ G.chromaticCardinal = ℵ_ 2 ∧
      ∀ (H : G.Subgraph) (_ : #H.verts = ℵ₁), H.coe.chromaticCardinal = ℵ₀ := by
…
```
theorem Erdos918.erdos_918.variants.eq_aleph_0.parts.ii In [Er69b] the questions are stated with $= \aleph_0$ rather than $\leq\aleph_0$ sorry line 130 source
Source documentation
In [Er69b] the questions are stated with $= \aleph_0$ rather than $\leq\aleph_0$. This is a likely typo since it can be shown that no such graph exists in this case.
This is the second question with induced subgraphs.

depends on sorry what this means
Statement uses: Not, Exists, SimpleGraph, And, Eq, Cardinal, Cardinal.mk, OrderEmbedding, Ordinal, Cardinal.aleph, HAdd.hAdd, Ordinal.omega0, OfNat.ofNat, SimpleGraph.chromaticCardinal, Set, SimpleGraph.induce, Cardinal.aleph0
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_918.variants.eq_aleph_0.parts.ii :
    ¬∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ (ω + 1) ∧ G.chromaticCardinal = ℵ₁ ∧
      ∀ (W : Set V) (_ : #W = ℵ_ ω), (G.induce W).chromaticCardinal = ℵ₀ := by
…
```
theorem Erdos918.erdos_918.variants.eq_aleph_0_all_subgraphs.parts.ii In [Er69b] the questions are stated with $= \aleph_0$ rather than $\leq\aleph_0$ sorry line 140 source
Source documentation
In [Er69b] the questions are stated with $= \aleph_0$ rather than $\leq\aleph_0$. This is a likely typo since it can be shown that no such graph exists in this case.
This is the second question with all subgraphs.

depends on sorry what this means
Statement uses: Not, Exists, SimpleGraph, And, Eq, Cardinal, Cardinal.mk, OrderEmbedding, Ordinal, Cardinal.aleph, HAdd.hAdd, Ordinal.omega0, OfNat.ofNat, SimpleGraph.chromaticCardinal, SimpleGraph.Subgraph, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, Cardinal.aleph0
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_918.variants.eq_aleph_0_all_subgraphs.parts.ii :
    ¬∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ (ω + 1) ∧ G.chromaticCardinal = ℵ₁ ∧
      ∀ (H : G.Subgraph) (_ : #H.verts = ℵ_ ω), H.coe.chromaticCardinal = ℵ₀ := by
…
```