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Statement

Setting (pp. 5--6). A family $\mathcal F\subseteq\mathcal H(\mathbb C)$ of entire functions is a Wetzel family when, for every $z\in\mathbb C$, the set $\{f(z):f\in\mathcal F\}$ has cardinality less than $|\mathcal F|$ (Definition 3.1, p. 5). A set $Y\subseteq\mathbb C$ with $|Y|<2^{\aleph_0}$ is universal (for entire functions) when for every $X\subseteq\mathbb C$ with $|X|<2^{\aleph_0}$ there is a non-constant entire function $f$ with $f(X)\subseteq Y$ (Definition 3.4, p. 6).

Proposition 3.7 (p. 6, quoted). "If there is a universal set there is also a Wetzel family."

Two companion statements on p. 6 frame it. Erdős's theorem, Proposition 3.5, says that under CH every countable dense set is universal, so Proposition 3.7 gives Corollary 3.8 (p. 7), Erdős's Wetzel family under CH. Proposition 3.6 says that a universal set $Y$ satisfies $|Y|^+=2^{\aleph_0}$, so the continuum is then a successor cardinal. The converse of Proposition 3.7 fails (Corollary 6.6, on the Theorem 6.5 page).

Proof pointer

Pp. 6--7. Enumerate $\mathbb C$ as $\langle z_\alpha:\alpha<\kappa\rangle$ with $\kappa=2^{\aleph_0}$, and for each $\alpha$ choose a non-constant entire $f_\alpha$ mapping $\{z_\beta:\beta<\alpha\}$ into $Y$. At $z_\alpha$ the family takes values in $Y$ or among the fewer than $\kappa$ values $f_\beta(z_\alpha)$ with $\beta\le\alpha$. The family has $\kappa$ members because $\kappa$ is regular, by Proposition 3.6, and a non-constant entire function cannot map all of $\mathbb C$ into $Y$.

Dependencies

Proposition 3.6 (p. 6) and the identity theorem (Proposition 2.1, p. 4).

Read depth

Claims checked: the definitions and the statement were read clause by clause on the printed pages. Nothing here is independently reviewed.

Source. Jonathan Schilhan and Thilo Weinert, Wetzel families and the continuum, J. Lond. Math. Soc. (2) 109 (2024), no. 6, Paper No. e12918, doi:10.1112/jlms.12918; arXiv:2310.19473. Labels and pages here are those of arXiv:2310.19473v3, the edition read, named on the source card.

Bears on

• Problem 1119: the paper records Kumar and Shelah's observation that a set of cardinality $\aleph_1$ universal for sets of cardinality $\aleph_1$, in a model of $2^{\aleph_0}=\aleph_2$, would give a Wetzel family there (p. 3); Proposition 3.7 is that implication for every value of the continuum. A Wetzel family in a model of $2^{\aleph_0}=\aleph_2$ is a family of more than $\aleph_1$ entire functions taking at most $\aleph_1$ values at each point. The proposition is a ZFC implication and decides nothing about the problem without a universal set; the paper produces one in Theorem 7.1.