Erdos 1964 interpolation problem associated continuum hypothesis

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generalization_p10: Erdős's generalization of the first part of his theorem: for cardinals n < m < c, a family of analytic functions with at most n distinct values at each point has power at most n.

question_p10: The paper's closing question asks whether there is a family of c distinct entire functions whose set of values at every point has power less than c, which Erdős says his construction gives when c = aleph_1, while for c > aleph_1 his proof breaks down.

theorem_p9: Erdős's theorem that if the continuum exceeds aleph_1 every family of analytic functions taking countably many values at each point is denumerable, while if the continuum equals aleph_1 some such family has the power of the continuum.

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P. Erdős: An interpolation problem associated with the continuum hypothesis, Michigan Math. J. 11 (1964), 9--10, doi:10.1307/mmj/1028999028; MR 29 #5744; Zentralblatt 121,258. No notice is printed on either page of the two-page scan; the hosting archive's site footer speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, read: "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only."); the journal's host Project Euclid could not be read on 2026-10-02, returning only a bot-detection page, and Crossref records no license for the DOI; the term is unstated.

Erdős answers a question of Wetzel from the Ann Arbor Problem Book: if a family {f_alpha} of analytic functions has property P_0, that for each z the set of values {f_alpha(z)} is countable, must the family be countable? His Theorem (p. 9) shows the answer depends on the continuum hypothesis: if c > aleph_1 then every family with P_0 is denumerable, while if c = aleph_1 some family with P_0 has power c; he reports being informed that R. D. Dixon proved the first part "last year" (the paper was received September 18, 1963). The positive half is a short counting argument — the union of the aleph_1 countable coincidence sets S(alpha,beta) has power at most aleph_1, so a point z_0 outside it separates all the functions; the negative half builds aleph_1 distinct entire functions by transfinite induction, each of the form f_gamma(z) = eps_0 + sum eps_n prod (z - w_i) with rapidly decreasing eps_n chosen to hit a fixed dense denumerable set S at prescribed points (p. 10). He notes the counting argument generalizes (p. 10): for cardinals n < m < c, if each set {f_alpha(z)} has at most n distinct values, the family has power at most n (equivalently, the case n^+ < c). Bearing on Problem 1119, this generalization answers it yes when the problem's bound, n here, satisfies n^+ < c, and the closing paragraph asks the related question without a uniform bound: Erdős says he is unable to decide whether there is a family of c distinct entire functions for which every value set {f_alpha(z_0)} has power less than c, the construction working for c = aleph_1 but the proof breaking down for c > aleph_1, and he points to Cohen's independence proof as making the question more interesting.

Source: https://users.renyi.hu/~p_erdos/1964-04.pdf.

Bears on. #1119: the remark on p. 10, read with the problem's $\mathfrak m$ as its $\mathfrak n$ and entire functions as analytic ones, answers the problem yes for every $\mathfrak m$ with $\mathfrak m^+<\mathfrak c$, in ZFC; it says nothing about the case $\mathfrak m^+=\mathfrak c$. The closing question (p. 10), which the paper leaves undecided, asks for $\mathfrak c$ distinct entire functions with fewer than $\mathfrak c$ values at each point; when $\mathfrak c=\mathfrak m^+$ such a family would answer the problem no at $\mathfrak m$.

Results.

• Theorem (p. 9): if $\mathfrak c>\aleph_1$ every family of analytic functions with property $P_0$ is denumerable; if $\mathfrak c=\aleph_1$ some such family has power $\mathfrak c$.
• Remark (p. 10): for cardinals $\mathfrak n<\mathfrak m<\mathfrak c$, a family of analytic functions with at most $\mathfrak n$ distinct values at each point has power at most $\mathfrak n$.
• Closing question (p. 10): can one construct $\mathfrak c$ distinct entire functions whose value set at every point has power less than $\mathfrak c$? Possible for $\mathfrak c=\aleph_1$, undecided for $\mathfrak c>\aleph_1$.

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