Erdős problem 1119 · problems.science

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

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FormalConjectures/ErdosProblems/

1119.lean

Retained formal statement3 of 5

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

FormalConjectures/ErdosProblems/1119.leanErdos1119.erdos_1119.variants.easy_case5 linesExact file

1∀ (m : Cardinal.{0}),2  Cardinal.aleph0 < m →3    Order.succ m < Cardinal.continuum →4      ∀ (F : Set (ℂ → ℂ)),5        (∀ f ∈ F, Differentiable ℂ f) → (∀ (z₀ : ℂ), Cardinal.mk ↑{y | ∃ f ∈ F, f z₀ = y} ≤ m) → Cardinal.mk ↑F ≤ m

SolvedStatement only, no proof

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