Problem 1119

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claims/: The 3 claim pages of Problem 1119, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $\mathfrak{m}$?

Formulation. Read as the site words it, the statement holds vacuously under CH, where no $\mathfrak m$ satisfies $\aleph_0<\mathfrak m<\mathfrak c$. In ZFC it holds for every $\mathfrak m$ with $\mathfrak m^+<\mathfrak c$, by Erdős's counting argument of 1964. Hayman calls that case easy, and the site records the independence of the remaining case $\mathfrak m^+=\mathfrak c$, which needs CH to fail. The standing answers the site's wording; it and the case $\mathfrak m^+=\mathfrak c$ are both independent of ZFC.

Status. Independent. The site's commentary records that the question is undecidable when $\mathfrak m^+=\mathfrak c$, crediting Kumar and Shelah with a model where the answer is yes and Schilhan and Weinert with a model where it is no, and that the answer is yes whenever $\mathfrak m^+<\mathfrak c$. The frontmatter standing is derived from the accepted claim page Schilhan and Weinert's result, which, together with any model of CH, where the statement holds vacuously, gives the independence of the statement in the site's wording, and, together with Kumar and Shelah's result, the independence of the case $\mathfrak m^+=\mathfrak c$; both are accepted on their refereed publication and the site's credit. The case $\mathfrak m^+<\mathfrak c$ is Erdős's 1964 result, an accepted partial claim.

Source. erdosproblems.com/1119, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1119, https://www.erdosproblems.com/1119.

References.

• [Er64g] Erdős, P., An interpolation problem associated with the continuum hypothesis. Michigan Math. J. (1964), 9-10.
• [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.
• [KuSh17] Kumar, Ashutosh and Shelah, Saharon, On a question about families of entire functions. Fund. Math. (2017), 279-288.
• [ScWe24] Schilhan, Jonathan and Weinert, Thilo, Wetzel families and the continuum. J. Lond. Math. Soc. (2) (2024), Paper No. e12918, 27.

Formalization. Statement in formal-conjectures. An outside Lean proof of the case $\mathfrak m^+<\mathfrak c$ is the formalization link on Erdős's claim page.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1964_interpolation_problem_associated_continuum_hypothesis
• erdos_1964_interpolation_problem_associated_continuum_hypothesis / generalization_p10
• erdos_1964_interpolation_problem_associated_continuum_hypothesis / question_p10
• erdos_1964_interpolation_problem_associated_continuum_hypothesis / theorem_p9
• kumar_2017_question_about_families_entire_functions
• kumar_2017_question_about_families_entire_functions / theorem_2_1
• kumar_2017_question_about_families_entire_functions / theorem_3_1
• schilhan_2024_wetzel_families_continuum
• schilhan_2024_wetzel_families_continuum / lemma_3_2
• schilhan_2024_wetzel_families_continuum / proposition_3_7
• schilhan_2024_wetzel_families_continuum / theorem_5_14
• schilhan_2024_wetzel_families_continuum / theorem_7_1