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Erdős problem 1167

Erdős Problem 1167. Let r≥2r \geq 2 be finite, γ≥2\gamma \geq 2, and λ\lambda be an infinite cardinal. Let κα\kappa_\alpha be cardinals for all α<γ\alpha < \gamma. Is it true that 2λ→(κα+1)α<γr+12^\lambda \to (\kappa_\alpha + 1)_{\alpha < \gamma}^{r+1} implies λ→(κα)α<γr?\lambda \to (\kappa_\alpha)_{\alpha < \gamma}^r? Here ++ means cardinal addition, so that κα+1=κα\kappa_\alpha + 1 = \kappa_\alpha if κα\kappa_\alpha is infinite.

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8 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

1167.lean

Retained formal statement7 of 8

The infinite_targets variant without the bound κα≤λ\kappa_\alpha \leq \lambda is false. Taking γ=1\gamma = 1, κ0=2ℵ0\kappa_0 = 2^{\aleph_0}, λ=ℵ0\lambda = \aleph_0: the premise 2ℵ0→(2ℵ0)132^{\aleph_0} \to (2^{\aleph_0})^3_1 holds since 2ℵ0≤2ℵ02^{\aleph_0} \leq 2^{\aleph_0}, but the conclusion ℵ0→(2ℵ0)12\aleph_0 \to (2^{\aleph_0})^2_1 fails by Cantor's theorem.

FormalConjectures/ErdosProblems/1167.leanErdos1167.erdos_1167.variants.infinite_targets_needs_bound7 linesExact file
¬∀ (r : ℕ),    2 ≤ r →      ∀ (lam : Cardinal.{u}),        Cardinal.aleph0 ≤ lam →          ∀ (γ : Ordinal.{u}) (κ : γ.ToType → Cardinal.{u}),            (∀ (i : γ.ToType), Cardinal.aleph0 ≤ κ i) →              Combinatorics.cardinalPartitionRel (2 ^ lam) (r + 1) γ κ → Combinatorics.cardinalPartitionRel lam r γ κ
TestStatement only, no proof