Source labels openChecked July 26, 2026

Erdős Problem 1175

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 $\kappa$ ?

Combinatorics AMS 5

Mathematical 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 $\kappa$ ?

Shelah proved that a negative answer is consistent when $\kappa = \lambda = \aleph_1$ (see erdos_1175.variants.shelah_consistency).

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_1175

Canonical source

Complete statement target, proof intentionally absentLean 4

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  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization erdosproblems.com/1175background