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 $\alpha_r \in O(r^5 \log^2 r)$ 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 $n/r + \alpha_r$.

翻译：2017年，Aharoni提出了Caccetta-H\"{a}ggkvist猜想的如下推广：若$G$是一个具有$n$个颜色类且每个颜色类大小至少为$r$的简单$n$顶点边着色图，则$G$包含一个长度至多为$\lceil n/r \rceil$的彩虹圈。本文证明，对于固定的$r$，Aharoni猜想在加法常数范围内成立。具体而言，我们证明对于每个固定的$r \geq 1$，存在常数$\alpha_r \in O(r^5 \log^2 r)$，使得若$G$是一个具有$n$个颜色类且每个颜色类大小至少为$r$的简单$n$顶点边着色图，则$G$包含一个长度至多为$n/r + \alpha_r$的彩虹圈。