Optimal quantitative global bound for rainbow perfect matchings

Determine whether every globally (n-o(n))-bounded colouring of the complete bipartite graph contains a rainbow perfect matching.

Background

The paper notes that its theorem establishes only qualitative boundedness and leaves the optimal quantitative constant unresolved. In the benchmark case of rainbow perfect matchings in complete bipartite graphs, existing results guarantee a rainbow perfect matching under a bound strictly below n on each colour class, while a bound of n is insufficient.

The unresolved issue is whether the threshold can be pushed asymptotically up to n, namely whether allowing each colour to occur on n-o(n) edges still forces a rainbow perfect matching.

References

From above, it is clear that a global bound of $n$ is necessary in order to have enough colours. It follows from a recent construction of Pokrovskiy and Sudakov (see also the very recent improvement ) that being globally $n$-bounded is not enough to guarantee a rainbow perfect matching, but it is still possible that a bound of the form $n-o(n)$ suffices.

— A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs  (2501.07644 - Kathapurkar et al., 13 Jan 2025) in Section 6, “Concluding Remarks”