{"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/new-bounds-for-ramsey-numbers-r-k-k-e-k-l-e","title":"New bounds for Ramsey numbers $R(K_k-e,K_l-e)$","arxiv_id":"2107.04460","date":"2021-07-09","proceeding":null,"authors":["Jan Goedgebeur","Steven Van Overberghe"],"abstract":"Let $R(H_1,H_2)$ denote the Ramsey number for the graphs $H_1, H_2$, and let $J_k$ be $K_k{-}e$. We present algorithms which enumerate all circulant and block-circulant Ramsey graphs for different types of graphs, thereby obtaining several new lower bounds on Ramsey numbers including: $49 \\leq R(K_3,J_{12})$, $36 \\leq R(J_4,K_8)$, $43 \\leq R(J_4,J_{10})$, $52 \\leq R(K_4,J_8)$, $37 \\leq R(J_5,J_6)$, $43 \\leq R(J_5,K_6)$, $65\\leq R(J_5,J_7)$. We also use a gluing strategy to derive a new upper bound on $R(J_5,J_6)$. With both strategies combined, we prove the value of two Ramsey numbers: $R(J_5,J_6)=37$ and $R(J_5,J_7)=65$. We also show that the 64-vertex extremal Ramsey graph for $R(J_5,J_7)$ is unique. Furthermore, our algorithms also allow to establish new lower bounds and exact values on Ramsey numbers involving wheel graphs and complete bipartite graphs, including: $R(W_7,W_4) = 21$, $R(W_7,W_7) = 19$, $R(K_{3,4},K_{3,4}) = 25$, and $R(K_{3,5}, K_{3,5})=33$.","url_abs":"https://arxiv.org/abs/2107.04460v1","url_pdf":"https://arxiv.org/pdf/2107.04460v1.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":"new-bounds-for-ramsey-numbers-r-k-k-e-k-l-e","repo_url":"https://github.com/Steven-VO/circulant-Ramsey","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}