Let $G_n$ be the coprime graph on $\{1,\ldots,n\}$. We prove that the mixed vertex-coloring coprime Ramsey number satisfies \[ \Rcop(k_1,\ldots,k_c)=p_{\sum_{i=1}^c(k_i-1)}, \] where $p_m$ is the $m$-th prime. The proof is elementary: the prime clique $\{1\}\cup\{p\le n:p\text{ prime}\}$ gives the upper bound by pigeonhole, while a prime-bin partition gives the matching lower bound by coloring each composite with a bin containing one of its prime divisors. We reserve $\Rcop$ for this vertex-coloring parameter; the edge-coloring parameter on the same host graph is denoted $\Redge$. The same certificate viewpoint yields several extensions, including a support-disjointness generalization, a polynomial-time certificate-extraction primitive, and an exact reduction of the edge-coloring variant to classical Ramsey numbers: $\Redge(k_1,\ldots,k_c)=p_{\Rcl(k_1,\ldots,k_c)-1}$. These two formulas are rank transfers from the same clique-label certificate. We also prove that the balanced two-color diagonal threshold equals the unrestricted threshold $p_{2k-2}$ for all $k\ge2$, via a deterministic prime-bin split requiring only the weak inequality $2p_m<p_{2m}<3p_m$; for fixed $c$, a Hall argument plus a standard Selberg--Delange estimate gives eventual multicolor balanced certificates.

翻译：设$G_n$为$\{1,\ldots,n\}$上的互素图。我们证明混合顶点染色互素拉姆齐数满足\[ \Rcop(k_1,\ldots,k_c)=p_{\sum_{i=1}^c(k_i-1)}, \]其中$p_m$是第$m$个素数。证明是初等的：素数团$\{1\}\cup\{p\le n:p\text{是素数}\}$通过鸽巢原理给出上界，而素数仓划分通过用含有一个素数因子的桶对每个合数染色，给出匹配的下界。我们将$\Rcop$保留用于此顶点染色参数；同一宿主图上的边染色参数记为$\Redge$。相同的证书视角导出了若干扩展，包括支撑不相交泛化、多项式时间证书提取原语，以及边染色变体到经典拉姆齐数的精确归约：$\Redge(k_1,\ldots,k_c)=p_{\Rcl(k_1,\ldots,k_c)-1}$。这两个公式均来自同一团标签证书的秩转移。我们还证明，对于所有$k\ge2$，平衡二色对角阈值等于无限制阈值$p_{2k-2}$，通过确定性素数仓分割仅需弱不等式$2p_m<p_{2m}<3p_m$；对于固定的$c$，霍尔论证加标准塞尔伯格-德兰热估计给出了最终的多色平衡证书。