1175.lean`

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5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 1175

Reference: erdosproblems.com/1175

[KoSh88] Komjáth, Péter and Shelah, Saharon, Forcing constructions for uncountably chromatic graphs. J. Symbolic Logic (1988), 696--707.

Formalization notes

Chromatic cardinal: SimpleGraph.chromaticCardinal is the cardinal-valued chromatic number defined in FormalConjecturesForMathlib. It extends the finite chromaticNumber (which takes values in ℕ∞) to a Cardinal, and is therefore able to distinguish between different infinite chromatic numbers.
Triangle-free subgraph: a subgraph H : G.Subgraph is triangle-free when H.coe.CliqueFree 3. This is the standard Mathlib formulation: CliqueFree 3 means the graph has no K₃ as a clique.
Subgraph: we use G.Subgraph (an arbitrary subgraph record) rather than an induced subgraph since the problem asks for any subgraph, not just induced ones.

Imports (1)

FormalConjecturesUtil

theorem Erdos1175.erdos_1175 Let $\kappa$ be an uncountable cardinal sorry line 55 source
Source documentation
Let $\kappa$ be an uncountable cardinal. Must there exist a cardinal $\lambda$ such that every graph with chromatic number $\lambda$ contains a triangle-free subgraph with chromatic number $\kappa$?
Shelah proved that a negative answer is consistent when $\kappa = \lambda = \aleph_1$ (see erdos_1175.variants.aleph_one).

depends on sorry what this means
Statement uses: Iff, True, Cardinal, LT.lt, Cardinal.aleph0, Exists, SimpleGraph, Eq, SimpleGraph.chromaticCardinal, SimpleGraph.Subgraph, And, SimpleGraph.CliqueFree, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1175 : answer(sorry) ↔
    ∀ (κ : Cardinal), ℵ₀ < κ →
      ∃ (μ : Cardinal),
        ∀ (V : Type*) (G : SimpleGraph V), G.chromaticCardinal = μ →
          ∃ (H : G.Subgraph), H.coe.CliqueFree 3 ∧ H.coe.chromaticCardinal = κ := by
…
```
theorem Erdos1175.erdos_1175.variants.aleph_one The case $\kappa = \lambda = \aleph_1$ of Problem 1175: does every graph with chromatic number $\aleph_1$ contain a triangle-free subgraph with chromatic number $\aleph_1$? sorry line 74 source
Source documentation
The case $\kappa = \lambda = \aleph_1$ of Problem 1175: does every graph with chromatic number $\aleph_1$ contain a triangle-free subgraph with chromatic number $\aleph_1$?
Shelah proved that a negative answer is consistent with ZFC [KoSh88, Theorem 1]: there is a model of ZFC containing a graph with chromatic number $\aleph_1$ all of whose triangle-free subgraphs have countable chromatic number. Consistency results about models of ZFC are not directly formalizable inside Lean's fixed model, so only the underlying question is recorded here. Note that this concerns only the choice $\lambda = \aleph_1$; it does not rule out a larger $\lambda$ in erdos_1175.

depends on sorry what this means
Statement uses: Iff, True, SimpleGraph, Eq, Cardinal, SimpleGraph.chromaticCardinal, OrderEmbedding, Ordinal, Cardinal.aleph, OfNat.ofNat, Exists, SimpleGraph.Subgraph, And, SimpleGraph.CliqueFree, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, Nat
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_1175.variants.aleph_one : answer(sorry) ↔
    ∀ (V : Type*) (G : SimpleGraph V), G.chromaticCardinal = ℵ_ 1 →
      ∃ (H : G.Subgraph), H.coe.CliqueFree 3 ∧ H.coe.chromaticCardinal = ℵ_ 1 := by
…
```
theorem Erdos1175.erdos_1175.variants.threshold_formulation Threshold reformulation variant sorry line 86 source
Source documentation
Threshold reformulation variant. Replaces chromaticCardinal = λ in the hypothesis of erdos_1175 with λ ≤ chromaticCardinal (a graph of chromatic number ≥ λ has a triangle-free subgraph of chromatic number κ). This is a strengthening of erdos_1175 (see erdos_1175.test.threshold_implies_exact).

depends on sorry what this means
Statement uses: Iff, True, Cardinal, LT.lt, Cardinal.aleph0, Exists, SimpleGraph, LE.le, SimpleGraph.chromaticCardinal, SimpleGraph.Subgraph, And, SimpleGraph.CliqueFree, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, OfNat.ofNat, Nat, Eq
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_1175.variants.threshold_formulation : answer(sorry) ↔
    ∀ (κ : Cardinal), ℵ₀ < κ →
      ∃ (μ : Cardinal),
        ∀ (V : Type*) (G : SimpleGraph V), μ ≤ G.chromaticCardinal →
          ∃ (H : G.Subgraph), H.coe.CliqueFree 3 ∧ H.coe.chromaticCardinal = κ := by
…
```
theorem Erdos1175.erdos_1175.test.exists_triangle_free_subgraph Documented declaration: exists_triangle_free_subgraph line 99 source
Source documentation
Every graph has a triangle-free subgraph: the bottom subgraph (with no edges) witnesses triangle-freeness, so the existential ∃ H : G.Subgraph, H.coe.CliqueFree 3 in erdos_1175 is non-vacuous.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Exists, SimpleGraph.Subgraph, SimpleGraph.CliqueFree, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, OfNat.ofNat, Nat
Proof uses: Exists.intro, Bot.bot, of_eq_true, EmptyCollection.emptyCollection, Set, eq_true, SimpleGraph.cliqueFree_bot, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true
17 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1175.test.exists_triangle_free_subgraph
    (V : Type*) (G : SimpleGraph V) : ∃ H : G.Subgraph, H.coe.CliqueFree 3 :=
…
```
theorem Erdos1175.erdos_1175.test.threshold_implies_exact Documented declaration: threshold_implies_exact line 108 source
Source documentation
The threshold variant threshold_formulation is stronger than the exact-equality form erdos_1175: if every graph with chromaticCardinal ≥ μ has the desired triangle-free subgraph, then in particular every graph with chromaticCardinal = μ does too.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Cardinal, LT.lt, Cardinal.aleph0, Exists, SimpleGraph, LE.le, SimpleGraph.chromaticCardinal, SimpleGraph.Subgraph, And, SimpleGraph.CliqueFree, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, OfNat.ofNat, Nat, Eq
Proof uses: Exists.intro, Eq.ge
13 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1175.test.threshold_implies_exact :
    (∀ (κ : Cardinal.{0}), ℵ₀ < κ →
      ∃ (μ : Cardinal.{0}),
        ∀ (V : Type) (G : SimpleGraph V), μ ≤ G.chromaticCardinal →
          ∃ (H : G.Subgraph), H.coe.CliqueFree 3 ∧ H.coe.chromaticCardinal = κ) →
    (∀ (κ : Cardinal.{0}), ℵ₀ < κ →
      ∃ (μ : Cardinal.{0}),
        ∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = μ →
          ∃ (H : G.Subgraph), H.coe.CliqueFree 3 ∧ H.coe.chromaticCardinal = κ) := by
…
```