{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/verified-certificates-via-sat-and-computer","title":"Verified Certificates via SAT and Computer Algebra Systems for the Ramsey R(3, 8) and R(3, 9) Problems","arxiv_id":"2502.06055","date":"2025-02-09","proceeding":null,"authors":["Zhengyu Li","Conor Duggan","Curtis Bright","Vijay Ganesh"],"abstract":"The Ramsey problem $R(3, k)$ seeks to determine the smallest value of $n$ such that any red/blue edge coloring of the complete graph on $n$ vertices must either contain a blue triangle (3-clique) or a red clique of size $k$. Despite its significance, many previous computational results for the Ramsey $R(3, k)$ problem such as $R(3, 8)$ and $R(3, 9)$ lack formal verification. To address this issue, we use the software MathCheck to generate certificates for Ramsey problems $R(3, 8)$ and $R(3, 9)$ (and symmetrically $R(8, 3)$ and $R(9, 3)$) by integrating a Boolean satisfiability (SAT) solver with a computer algebra system (CAS). Our SAT+CAS approach significantly outperforms traditional SAT-only methods, demonstrating an improvement of several orders of magnitude in runtime. For instance, our SAT+CAS approach solves $R(3, 8)$ (resp., $R(8, 3)$) sequentially in 59 hours (resp., in 11 hours), while a SAT-only approach using state-of-the-art CaDiCaL solver times out after 7 days. Additionally, in order to be able to scale to harder Ramsey problems $R(3, 9)$ and $R(9, 3)$ we further optimized our SAT+CAS tool using a parallelized cube-and-conquer approach. Our results provide the first independently verifiable certificates for these Ramsey numbers, ensuring both correctness and completeness of the exhaustive search process of our SAT+CAS tool.","url_abs":"https://arxiv.org/abs/2502.06055v1","url_pdf":"https://arxiv.org/pdf/2502.06055v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"verified-certificates-via-sat-and-computer","repo_url":"https://github.com/ConDug/MathCheckRamsey","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2502.06055","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}