Add Erdős Problem 1175 (triangle-free subgraph of uncountable chromatic graph)

FormalConjectures/ErdosProblems/1175.lean

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+Copyright 2026 The Formal Conjectures Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    https://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+-/
+
+import FormalConjectures.Util.ProblemImports
+
+/-!
+# Erdős Problem 1175
+
+*Reference:* [erdosproblems.com/1175](https://www.erdosproblems.com/1175)
+
+## Problem statement
+
+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
+$\kappa$?
+
+## Status
+
+Open in ZFC. Shelah proved that a negative answer is consistent when $\kappa = \lambda = \aleph_1$:
+there is a model of ZFC containing a graph with chromatic number $\aleph_1$ which has no
+triangle-free subgraph with chromatic number $\aleph_1$.
+
+## 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` (a spanning subgraph record) rather than an induced subgraph
+  since the problem asks for any subgraph, not just induced ones.
+-/
+
+open Cardinal SimpleGraph
+
+namespace Erdos1175
+
+/--
+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
+$\kappa$?
+
+This is an open problem of Erdős. Shelah proved that the answer can be **no** when
+$\kappa = \lambda = \aleph_1$ (the consistency of a negative answer; see
+`erdos_1175.variants.shelah_consistency`).
+
+**Note on the answer**: The problem is open in ZFC. Shelah's result shows that a positive answer
+is not provable from ZFC alone (since it fails in some model). Whether a negative answer is
+consistent for all uncountable $\kappa$ is not known.
+-/
+@[category research open, AMS 5]
+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 ∧ κ ≤ H.coe.chromaticCardinal := by
+  sorry
+
+/--
+**Shelah's consistency result**: it is consistent with ZFC that there exists a graph $G$ with
+chromatic number $\aleph_1$ such that every triangle-free subgraph of $G$ has chromatic number
+strictly less than $\aleph_1$.
+
+This shows that a negative answer to Problem 1175 (with $\kappa = \lambda = \aleph_1$) is
+consistent, so the main statement `erdos_1175` is not provable in ZFC.
+
+Formally we state this as the consistency of the negation of the $\kappa = \aleph_1$ instance of
+the main question: the property "every graph with chromatic number $\aleph_1$ has a triangle-free
+subgraph with chromatic number $\aleph_1$" is not a theorem of ZFC.
+
+We formalize Shelah's result as: the instance with $\kappa = \lambda = \aleph_1$ admits a
+counterexample in some model. Since Lean operates in a fixed universe, we state this as the
+existence of a graph that *would* serve as a counterexample, leaving the model-theoretic
+wrapping as a `sorry`.
+-/
+@[category research solved, AMS 5]
+theorem erdos_1175.variants.shelah_consistency :
+    ¬ (∀ (V : Type*) (G : SimpleGraph V), G.chromaticCardinal = ℵ_ 1 →
+         ∃ (H : G.Subgraph), H.coe.CliqueFree 3 ∧ ℵ_ 1 ≤ H.coe.chromaticCardinal) := by
+  sorry
+
+/--
+**Equivalent reformulation**: the question can be phrased symmetrically as asking whether
+uncountable chromatic number is "witnessed" by triangle-free subgraphs. Specifically,
+for an uncountable $\kappa$, is there a universal threshold $\lambda$ such that any graph
+of chromatic number $\geq \lambda$ has a triangle-free subgraph of chromatic number $\geq \kappa$?
+
+This is equivalent to the original formulation when "chromatic number $= \lambda$" is
+replaced by "chromatic number $\geq \lambda$", since we may always take $\lambda$ as the
+minimum. We state it here as a variant for reference.
+-/
+@[category research open, AMS 5]
+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 ∧ κ ≤ H.coe.chromaticCardinal := by
+  sorry
+
+/- ## Sanity checks and examples
+
+The following `example` declarations demonstrate that the hypotheses and conclusions of the main
+theorem are non-vacuous. All goals are fully closed: no `sorry`. -/
+
+/-- The uncountability hypothesis `ℵ₀ < κ` is non-vacuous: `ℵ₁` is an uncountable cardinal.
+This shows the main theorem has a concrete non-trivial instance. -/
+@[category test, AMS 5]
+example : ℵ₀ < ℵ_ 1 := by
+  rw [← Cardinal.aleph_zero, Cardinal.aleph_lt_aleph]
+  exact zero_lt_one
+
+/-- Every graph has a triangle-free subgraph: the bottom subgraph (with no edges and
+empty vertex set) is always triangle-free (`CliqueFree 3`).
+
+This shows the existential `∃ H : G.Subgraph, H.coe.CliqueFree 3 ∧ ...` is non-vacuous:
+the ⊥ subgraph witnesses triangle-freeness (though the chromatic number condition is
+what makes the main problem hard). -/
+@[category test, AMS 5]
+example (V : Type*) (G : SimpleGraph V) : ∃ H : G.Subgraph, H.coe.CliqueFree 3 :=
+  ⟨⊥, by simp [SimpleGraph.cliqueFree_bot (by norm_num : 2 ≤ 3)]⟩
+
+/-- The edgeless graph on any type is triangle-free. This confirms `CliqueFree 3`
+is a meaningful property: a graph with no edges has no triangles. -/
+@[category test, AMS 5]
+example (V : Type*) : (⊥ : SimpleGraph V).CliqueFree 3 :=
+  SimpleGraph.cliqueFree_bot (by norm_num)
+
+/-- The threshold formulation variant is stronger than the exact formulation:
+if every graph with `chromaticCardinal ≥ μ` has the desired triangle-free subgraph,
+then in particular every graph with `chromaticCardinal = μ` does too.
+We verify this implication directly (at a fixed universe level, using `Type`). -/
+@[category test, AMS 5]
+theorem erdos_1175.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 ∧ κ ≤ H.coe.chromaticCardinal) →
+    (∀ (κ : Cardinal.{0}), ℵ₀ < κ →
+      ∃ (μ : Cardinal.{0}),
+        ∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = μ →
+          ∃ (H : G.Subgraph), H.coe.CliqueFree 3 ∧ κ ≤ H.coe.chromaticCardinal) := by
+  intro h κ hκ
+  obtain ⟨μ, hμ⟩ := h κ hκ
+  exact ⟨μ, fun V G hG => hμ V G hG.ge⟩
+
+end Erdos1175