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We show that sets avoiding 6-term arithmetic progressions in $\mathbb{Z}_6^n$have size at most $5.709^n$. It is also pointed out that the "productconstruction" does not work in this setting, specially, we show that for theextremal sizes in small dimensions we have $r_6(\mathbb{Z}_6)=5$,$r_6(\mathbb{Z}_6^2)=25$ and $ 116\leq r_6(\mathbb{Z}_6^3)\leq 124$.

Sets avoiding six-term arithmetic progressions in $\mathbb{Z}_6^n$ are exponentially small