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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 statement3 of 5

The 'easy' case of Erdős Problem 1119: if moreover m+<c\mathfrak{m}^+ < \mathfrak{c}, then any family of entire functions taking at most m\mathfrak{m} distinct values at each point has cardinality at most m\mathfrak{m}. In [Ha74] it is written that this is 'easy to see'.

FormalConjectures/ErdosProblems/1119.leanErdos1119.erdos_1119.variants.easy_case5 linesExact file
∀ (m : Cardinal.{0}),  Cardinal.aleph0 < m →    Order.succ m < Cardinal.continuum →      ∀ (F : Set (ℂ → ℂ)),        (∀ f ∈ F, Differentiable ℂ f) → (∀ (z₀ : ℂ), Cardinal.mk ↑{y | ∃ f ∈ F, f z₀ = y} ≤ m) → Cardinal.mk ↑F ≤ m
SolvedStatement only, no proof