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 $\lceil n/r\rceil$.

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 $r⩾1$, 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 $n/r+{\alpha }_r$.