Prime Certificates for Exact Vertex-Coprime Ramsey Numbers

2026-05-26

AI Summary

This study investigates coprime Ramsey numbers under vertex and edge colorings, establishing a precise connection with the distribution of prime numbers. By introducing prime bucket partitions, constructing prime-based cliques, and employing elementary combinatorial tools—including the pigeonhole principle, Hall’s theorem, and the Selberg–Delange estimate—the authors derive explicit formulas for the first time: in the vertex case, $R_{\mathrm{cop}}(k_1,\dots,k_c) = p_{\sum (k_i - 1)}$, and in the edge case, $R_{\mathrm{edge}}(k_1,\dots,k_c) = p_{R_{\mathrm{cl}}(k_1,\dots,k_c) - 1}$. The work introduces the novel perspective of “prime certificates,” enabling an exact reduction from coprime Ramsey numbers to classical Ramsey numbers, and proves that the balanced two-color threshold coincides with the unrestricted threshold $p_{2k-2}$.

Abstract

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 three extensions: 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.

Problem

• coprime Ramsey numbers
• vertex-coloring
• prime certificates
• Ramsey theory
• graph coloring

Innovation

• prime certificates
• coprime Ramsey numbers
• prime-bin partition
• rank transfer
• polynomial-time certificate extraction