  In 2017, Aharoni proposed the following generalization of the
Caccetta-H\"{a}ggkvist conjecture: if $G$ is a simple $n$-vertex edge-colored
graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow
cycle of length at most $\lceil n/r \rceil$.
  In this paper, we prove that, for fixed $r$, Aharoni's conjecture holds up to
an additive constant. Specifically, we show that for each fixed $r \geq 1$,
there exists a constant $c_r$ such that if $G$ is a simple $n$-vertex
edge-colored graph with $n$ color classes of size at least $r$, then $G$
contains a rainbow cycle of length at most $n/r + c_r$.
