Exact zero-sum Ramsey number of complete graphs in the 7 modulo 9 case

Determine, for every integer n congruent to 7 modulo 9, the exact value of the zero-sum Ramsey number R(K_n, Z_3), which is either n+3 or n+4.

Background

The paper surveys prior work on zero-sum Ramsey numbers modulo 3 and explains that the values of R(K_n, Z_3) are not completely determined. The unresolved case concerns complete graphs whose order is congruent to 7 modulo 9; existing results narrow the value to two possibilities, n+3 and n+4.

References

Determine for $n \equiv 7 \ (!!!!!!!!\mod 9)$ the exact value of $R(K_n, \mathbb{Z}_3)$, which is either $n + 3$ or $n + 4$.

— On zero-sum Ramsey numbers modulo 3  (2502.03864 - Caro et al., 6 Feb 2025) in Section 1, subsection “Determining R(G, Z_3),” Problem