Determine the exact mathbb{Z}_3-Ramsey number of complete graphs in the unresolved congruence case

Determine the exact value of the mathbb{Z}_3-Ramsey number R(K_n,mathbb{Z}_3) when n is congruent to 7 modulo 9, resolving whether it equals n+3 or n+4.

Background

The paper places its forest result within the broader program of determining zero-sum Ramsey numbers over mathbb{Z}_3. Although the authors mention the general problem for all graphs, they identify a specific unresolved case for complete graphs: when the number of vertices is congruent to 7 modulo 9, existing results determine only two possible values, n+3 and n+4.

Resolving this problem would settle the exact zero-sum Ramsey number for an infinite congruence class of complete graphs and provide progress toward the broader characterization sought for mathbb{Z}_3.

References

We note that this is far from being solved, as, for instance, the exact value of $R(K_n,\mathbb{Z}_3)$ is not know if $n \equiv 7 \pmod 9$ (it is known to be either $n+3$ or $n+4$).

— On a problem of Caro on $\mathbb{Z}_3$-Ramsey number of forests  (2503.01032 - Alvarado et al., 2 Mar 2025) in Section "Open questions" (Section 5)