On a problem concerning $\mathbb Z_3$-Ramsey numbers

For a graph $G$ with $3\mid e(G)$, let $R(G,\mathbb Z_3)$ denote the least integer $N$ such that every edge weighting $w:E(K_N)\to\mathbb Z_3$ admits a copy of $G$ whose edge weights sum to zero. We prove that every $n$-vertex graph $G$ with $n\ge6$ and $3\mid e(G)$ satisfies $R(G,\mathbb Z_3)\le n+