Aharoni's rainbow cycle conjecture holds up to an additive constant

In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist 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 $\lceil n\mathrm{/}r\rceil$. 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 $r\ge 1$, 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 $n\mathrm{/}r+{\alpha }_r$.