  We provide answers to a question brought up by Erd\H{o}s about the
construction of Wetzel families in the absence of the continuum hypothesis - a
Wetzel family is a family $\mathcal{F}$ of entire functions on the complex
plane which pointwise assumes fewer than $\vert \mathcal{F} \vert$ values. To
be more precise, we show that the existence of a Wetzel family is consistent
with all possible values $\kappa$ of the continuum and, if $\kappa$ is regular,
also with Martin's Axiom. In the particular case of $\kappa = \aleph_2$ this
answers an open question asked by Kumar and Shelah. In the buildup to this
result, we are also solving an open question of Zapletal on strongly almost
disjoint functions. We also study a strongly related notion of sets exhibiting
a universality property via mappings by entire functions and show that these
consistently exist while the continuum equals $\aleph_2$.
