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Statement

Question (p. 10, unnumbered, quoted). "Can one construct a family of distinct entire functions $f_\alpha$ $(1\leq\alpha<\Omega_{\mathfrak c})$ such that for every $z$ the set $\{f_\alpha(z)\}$ has power less than $\mathfrak c$?"

Erdős says he is unable to decide it. He notes that the construction is possible if $\mathfrak c=\aleph_1$, by the second part of the theorem, whose value sets are countable, and that for $\mathfrak c>\aleph_1$ his proof breaks down. He adds that Cohen's recent proof of the independence of the continuum hypothesis gives the problem added interest.

Source. P. Erdős, An interpolation problem associated with the continuum hypothesis, Michigan Math. J. 11 (1964), 9--10, doi:10.1307/mmj/1028999028, p. 10; the edition read is named on the source card.

Read depth. Claims checked: the question was read clause by clause on the page image of p. 10. Nothing here is independently reviewed.

Proof pointer

None: the paper poses the question without an answer.

Dependencies

The theorem (p. 9), for the case $\mathfrak c=\aleph_1$.

Bears on

• Problem 1119: the question bounds each value set below $\mathfrak c$ rather than by a fixed cardinal. When $\mathfrak c=\mathfrak m^+$ with $\mathfrak m>\aleph_0$, power less than $\mathfrak c$ means at most $\mathfrak m$, and a family as asked is a family of more than $\mathfrak m$ entire functions with at most $\mathfrak m$ values at each point, the problem's case $\mathfrak m^+=\mathfrak c$ answered no. The paper decides nothing in that case; the answers are on the problem page.