  For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of
out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that
for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v)
\ge k$ for all $v \in V(G)$, then $G$ contains a directed cycle of length at
most $\lceil n/k \rceil$. Aharoni proposed a generalization of this conjecture,
that a simple edge-colored graph on $n$ vertices with $n$ color classes, each
of size $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. With
Pelik\'anov\'a and Pokorn\'a, we showed that this conjecture is true if each
color class has size ${\Omega}(k\log k)$. In this paper, we present a proof of
the conjecture if each color class has size ${\Omega}(k)$, which improved the
previous result and is only a constant factor away from Aharoni's conjecture.
We also consider what happens when the condition on the number of colors is
relaxed.
