Source labels openChecked July 26, 2026

Erdős Problem 70

Erdős Problem 70*: Let $\mathfrak{c}$ be the cardinality of the continuum, let $\beta$ be a countable ordinal, and let $2 \le n < \omega$ . Is it true that $\mathfrak{c} \to (\beta, n)^3_2$ ?

Mathematical logic AMS 3

Mathematical statement

Erdős Problem 70*: Let $\mathfrak{c}$ be the cardinality of the continuum, let $\beta$ be a countable ordinal, and let $2 \le n < \omega$ . Is it true that $\mathfrak{c} \to (\beta, n)^3_2$ ?

Note: The cases $n \le 3$ are trivially true (see omega_three), so the genuine content of the conjecture begins at $n = 4$ .

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_70

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_70 :    answer(sorry) ↔    ∀ᵉ (β : Ordinal.{0}) (n : ℕ) (_ : β.card ≤ ℵ₀) (_ : 2 ≤ n),      OrdinalCardinalRamsey3 (𝔠).ord β n := 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/70background