Universal upper bound for zero-sum Ramsey numbers modulo 3

Determine whether every graph G on n vertices with 3 dividing e(G) satisfies R(G, Z_3)≤n+8, with equality if and only if G=K_3.

Background

The paper identifies a general unresolved question for graphs with a number of edges divisible by 3. It asks whether a uniform upper bound of n+8 holds for all such graphs and whether the triangle is the unique extremal graph attaining equality.

References

Let $G$ be a graph $n$ vertices such that $3 \vert e(G)$. Is $R(G, \mathbb{Z}_3) \leq n + 8$ with equality if, and only if, $G = K_3$?

— 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