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Erdős problem 1119

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Let m\mathfrak{m} be an infinite cardinal with ℵ0<m<c=2ℵ0\aleph_0 < \mathfrak{m} < \mathfrak{c} = 2^{\aleph_0}. Let {fα}\{f_\alpha\} be a family of entire functions such that, for every z0∈Cz_0 \in \mathbb{C}, there are at most m\mathfrak{m} distinct values of fα(z0)f_\alpha(z_0). Must {fα}\{f_\alpha\} have cardinality at most m\mathfrak{m}?

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FormalConjectures/ErdosProblems/

1119.lean

Retained formal statement4 of 5

Erdős's theorem [Er64g], answering a question of Wetzel: if c>ℵ1\mathfrak{c} > \aleph_1, then every family of entire functions taking only countably many distinct values at each point z0∈Cz_0 \in \mathbb{C} is itself countable.

FormalConjectures/ErdosProblems/1119.leanErdos1119.erdos_1119.variants.erdos_wetzel2 linesExact file
Cardinal.aleph 1 < Cardinal.continuum →  ∀ (F : Set (ℂ → ℂ)), (∀ f ∈ F, Differentiable ℂ f) → (∀ (z₀ : ℂ), {y | ∃ f ∈ F, f z₀ = y}.Countable) → F.Countable
SolvedStatement only, no proof