---

Statement

Setting (pp. 4--5). $\mathcal H(\mathbb C)$ is the set of entire functions. A family $\mathcal F\subseteq\mathcal H(\mathbb C)$ 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).

Lemma 3.2 (p. 5). In ZFC, let $\mathcal F$ be a Wetzel family. Then:

(1) $|\mathcal F|=2^{\aleph_0}$;

(2) for every $\lambda<2^{\aleph_0}$, the set of $z\in\mathbb C$ with $|\{f(z):f\in\mathcal F\}|<\lambda$ has cardinality less than $2^{\aleph_0}$.

The paper cites (1) in the introduction: the Kumar and Shelah model with a Wetzel family of cardinality $\aleph_{\omega_1}$ has continuum $\aleph_{\omega_1}$ (p. 3).

Proof pointer

P. 5. Two distinct entire functions agree only on a countable set, by the identity theorem (Proposition 2.1, p. 4). If $|\mathcal F|<2^{\aleph_0}$, the union of these agreement sets over pairs from $\mathcal F$ has fewer than $2^{\aleph_0}$ points, and at a point outside it the family takes $|\mathcal F|$ distinct values, against the Wetzel property. Part (2) runs the same count on a subfamily of size $\lambda$.

Dependencies

The identity theorem (Proposition 2.1, p. 4).

Read depth

Claims checked: the definition and the statement were read clause by clause on the printed page. 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: suppose a family as in the problem has more than $\mathfrak m$ members. At each point it takes at most $\mathfrak m$ values, which is fewer than its number of members. So it is a Wetzel family, and by part (1) it has exactly $2^{\aleph_0}$ members. The paper does not discuss the problem's parameter $\mathfrak m$.