Wetzel families and the continuum

We provide answers to a question brought up by Erdős about the construction of Wetzel families in the absence of the continuum hypothesis: A Wetzel family is a family (Formula presented.) of entire functions on the complex plane which pointwise assumes fewer than (Formula presented.) values. To be more precise, we show that the existence of a Wetzel family is consistent with all possible values (Formula presented.) of the continuum and, if (Formula presented.) is regular, also with Martin's Axiom. In the particular case of (Formula presented.) this answers the main open question asked by Kumar and Shelah [Fund. Math. 239 (2017) no. 3, 279–288]. In the buildup to this result, we are also solving an open question of Zapletal on strongly almost disjoint functions from Zapletal [Israel J. Math. 97 (1997) no. 1, 101–111]. 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 (Formula presented.).