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Claim. For every cardinal $\kappa$ of uncountable cofinality there is, over a model of the generalized continuum hypothesis, a cardinal- and cofinality-preserving forcing extension with $\mathfrak c=\kappa$ containing a Wetzel family of size $\kappa$: a family of $\kappa$ entire functions that at every point takes fewer than $\kappa$ values. With $\kappa=\aleph_2$ the extension has $\mathfrak c=\aleph_2$ and a family of $\aleph_2$ entire functions taking at most $\aleph_1$ values at each point, so the question of Problem 1119 has a negative answer there for $\mathfrak m=\aleph_1$. This is Theorem 5.14 of arXiv:2310.19473v3 (13 May 2024), published as Jonathan Schilhan and Thilo Weinert, Wetzel families and the continuum, J. Lond. Math. Soc. (2) 109 (2024), no. 6, Paper No. e12918; the paper was first posted as arXiv:2310.19473 on 2023-10-30, which dates the page, and the numbering of the published version was not compared. The source card records the statements.

The negation of the problem's statement is therefore consistent with ZFC. The statement itself is consistent too, in two ways: under the continuum hypothesis no cardinal $\mathfrak m$ satisfies $\aleph_0<\mathfrak m<\mathfrak c$, so the statement holds vacuously in every model of CH; and in Kumar and Shelah's model, where CH fails, $\mathfrak c=\aleph_2$ and the answer for $\mathfrak m=\aleph_1$ is yes. So the statement is independent of ZFC, if ZFC is consistent: neither the statement nor its negation is provable. The statement is provable for every $\mathfrak m$ with $\mathfrak m^+<\mathfrak c$, by Erdős's counting argument of 1964 (source card), so the independence concerns the case $\mathfrak m^+=\mathfrak c$, the only case left open by Hayman's 1974 problem list; the independence of that case needs both this theorem and Kumar and Shelah's model, and the two models both have $\mathfrak c=\aleph_2$ and $\mathfrak m=\aleph_1$.

Depends on. The consistency of a positive answer in the case $\mathfrak m^+=\mathfrak c$ with CH failing is Kumar and Shelah's result; this page's own theorem supplies the consistency of a negative answer.

Argument, in outline. Section 3 proves in ZFC that every Wetzel family has size exactly $2^{\aleph_0}$ (Lemma 3.2) and that a set universal for sets of complex numbers under entire maps yields a Wetzel family (Proposition 3.7). Section 4 forces a family of strongly almost disjoint functions, and Section 5 builds the Wetzel family by a forcing that preserves cardinals and cofinalities, with Martin's Axiom also forceable when $\kappa$ is regular. The authors answer the question Kumar and Shelah left open, whether a Wetzel family is consistent with $\mathfrak c=\aleph_2$. The proof is not reconstructed on this page.

Acceptance. The result appeared in a refereed journal, the Journal of the London Mathematical Society, in 2024, the refereed evidence; the statements are those of the arXiv preprint, not compared with the published version. The site's curator, Thomas Bloom, marks the problem INDEPENDENT and records in the commentary that the question is undecidable when $\mathfrak m^+=\mathfrak c$, crediting Kumar and Shelah with the model where the answer is yes and Schilhan and Weinert with the model where it is no: that curator credit is the reviewed evidence. The independent claim value combines this page's theorem with the linked Kumar--Shelah page; the paper itself claims only the consistency of a negative answer. The formal-conjectures statement file for the problem, at the commit linked, holds statements only, every research declaration with a sorry body; its easy-case declaration points to an outside Lean proof of the case $\mathfrak m^+<\mathfrak c$ in Boris Alexeev's lean-proofs repository, whose header names Erdős as the informal author and Codex and GPT-5.6 Sol as the formal authors. Neither formalizes this result: the easy-case proof is the formalization link on Erdős's page, and this page lists no formalized evidence.