  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 at least $k$, has a rainbow cycle of length at most $\lceil n/k
\rceil$. Let us call $(\alpha, \beta)$ \emph{triangular} if every simple
edge-colored graph on $n$ vertices with at least $\alpha n$ color classes, each
with at least $\beta n$ edges, has a rainbow triangle. Aharoni, Holzman, and
DeVos showed the following: $(9/8,1/3)$ is triangular; $(1,2/5)$ is triangular.
In this paper, we improve those bounds, showing the following: $(1.1077,1/3)$
is triangular; $(1,0.3988)$ is triangular. Our methods give results for
infinitely many pairs $(\alpha, \beta)$, including $\beta < 1/3$; we show that
$(1.3481,1/4)$ is triangular.
