Grinblat's rainbow matching conjecture

Grinblat’s conjecture asserts that every admissibly colored multigraph with $n$ colors and each color spanning at least $3n-2$ vertices has a rainbow matching of size $n$. The conjecture is attributed to Grinblat and was posed roughly two decades ago.

Known results

• A construction using disjoint unions of $n-1$ triangles gives the lower threshold $3n-2$.
• A greedy argument gives the upper threshold $4n-3$.
• Grinblat, Nivasch and Omri, followed by Clemens, Ehrenmüller and Pokrovskiy, obtained asymptotic improvements, including $3n+O(n^{3/4})$.
• Munhá Correia and Sudakov proved the exact threshold $3n-2$ for all sufficiently large $n$.

April 2022 full-proof claim

An April 2022 arXiv entry states that Munhá Correia and Sudakov subsequently fully proved Grinblat’s conjecture. This conflicts with their earlier paper, which explicitly left small values $n\geq4$ open; the retrieved sources provide no independent verification resolving that discrepancy.

Current status (as of September 2026): A later source claims the conjecture is fully proved by Munhá Correia and Sudakov, but the available record contains an unresolved conflict with their earlier statement that small values $n\geq4$ remained open.