Erdős Problem #1119 — Families of Entire Functions with Few Pointwise Values

Let $\mathfrak m$ be a cardinal satisfying $\aleph_0<\mathfrak m<\mathfrak c$, where $\mathfrak c=2^{\aleph_0}$. Suppose $F$ is a family of entire functions $f:\mathbb C\to\mathbb C$ such that, for every $z_0\in\mathbb C$, the set $\{f(z_0):f\in F\}$ has cardinality at most $\mathfrak m$. Must $F$ itself have cardinality at most $\mathfrak m$?