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For any poset $P$, we show $\\operatorname{RT}(\\mathcal{B};n,P,l,t)\\le \\operatorname{RT}^{\\sharp}(\\mathcal{B};n,P,l,t)$, with equality when $P$ is a chain. In particular, for $t=1$, $\\operatorname{RT}(\\mathcal{B};n,C_k,l)=\\operatorname{RT}^{\\sharp}(\\mathcal{B};n,C_k,l)=(k-1)(l-1)$. We also give universal upper bounds for both versions. 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