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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).

Theorem 6.5 (p. 21, quoted). "$\mathsf{MA}+\neg\mathsf{CH}$ implies that there is no universal set."

Corollary 6.6 (p. 21, quoted). "The existence of a Wetzel family does not imply the existence of a universal set."

The corollary shows that the converse of Proposition 3.7 fails. Its proof takes $\kappa=\aleph_2$ in Theorem 5.14, whose extension has a Wetzel family and, $\aleph_2$ being regular, satisfies MA with $2^{\aleph_0}=\aleph_2$, so by Theorem 6.5 has no universal set. The paper adds that, granting a weakly inaccessible cardinal, the corollary already follows from Theorem 5.14 with Proposition 3.6 (p. 21).

Proof pointer

Section 6, pp. 19--21. The ccc poset $\mathbb S$ (p. 19) and Lemmas 6.1--6.4 are the tools. Given $Y$ with $|Y|<2^{\aleph_0}$, MA yields a set $Z$ with $Y\subseteq Z+(\mathbb Q+i\mathbb Q)$ whose pairwise distances lie in a union $U$ of rapidly shrinking intervals. Lemma 6.3 gives a set $X$ of size $\aleph_1$ whose pairwise distances lie in an open set $O$ of a different scale, and Lemma 6.4 shows that no non-constant entire function maps an uncountable set with distances in $O$ into a set with distances in $U$.

Dependencies

Lemmas 6.1--6.4 (pp. 19--21); for the corollary, Theorem 5.14 (p. 17).

Read depth

Claims checked: both statements and the proof of the corollary were read clause by clause on the printed page. The proof of Theorem 6.5 was not checked step by step. 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

None directly. The paper asks whether MA or PFA implies that a Wetzel family exists (Question 8.1, p. 24).