{"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/a-computerised-classification-of-some-almost","title":"A computerised classification of some almost minimal triangle-free Ramsey graphs","arxiv_id":"1710.06644","date":"2017-10-18","proceeding":null,"authors":["Oliver Krüger"],"abstract":"A graph $G$ is called a $(3,j;n)$-minimal Ramsey graph if it has the least amount of edges, $e(3,j;n)$, given that $G$ is triangle-free, the independence number $\\alpha(G) < j$ and that $G$ has $n$ vertices. Triangle-free graphs $G$ with $\\alpha(G) < j$ and where $e(G) - e(3,j;n)$ is small are said to be almost minimal Ramsey graphs. We look at a construction of some almost minimal Ramsey graphs, called $H_{13}$-patterned graphs. We make computer calculations of the number of almost minimal Ramsey triangle-free graphs that are $H_{13}$-patterned. The results of these calculations indicate that many of these graphs are in fact $H_{13}$-patterned. In particular, all but one of the connected $(3,j;n)$-minimal Ramsey graphs for $j \\leq 9$ are indeed $H_{13}$-patterned.","url_abs":"https://arxiv.org/abs/1710.06644v1","url_pdf":"https://arxiv.org/pdf/1710.06644v1.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":"a-computerised-classification-of-some-almost","repo_url":"https://github.com/tbk1988/almostminimalramsey","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}