{"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/online-size-ramsey-numbers-path-vs-c-4","title":"Online size Ramsey numbers: Path vs $C_4$","arxiv_id":"2211.12204","date":"2022-11-22","proceeding":null,"authors":["Grzegorz Adamski","Małgorzata Bednarska-Bzdęga"],"abstract":"Given two graphs $G$ and $H$, a size Ramsey game is played on the edge set of $K_\\mathbb{N}$. In every round, Builder selects an edge and Painter colours it red or blue. Builder's goal is to force Painter to create a red copy of $G$ or a blue copy of $H$ as soon as possible. The online (size) Ramsey number $\\tilde r(G,H)$ is the number of rounds in the game provided Builder and Painter play optimally. We prove that $\\tilde r(C_4,P_n)\\le 2n-2$ for every $n\\ge 8$. The upper bound matches the lower bound obtained by J. Cyman, T. Dzido, J. Lapinskas, and A. Lo, so we get $\\tilde r(C_4,P_n)=2n-2$ for $n\\ge 8$. Our proof for $n\\le 13$ is computer assisted. The bound $\\tilde r(C_4,P_n)\\le 2n-2$ solves also the \"all cycles vs. $P_n$\" game for $n\\ge 8$ $-$ it implies that it takes Builder $2n-2$ rounds to force Painter to create a blue path on $n$ vertices or any red cycle.","url_abs":"https://arxiv.org/abs/2211.12204v2","url_pdf":"https://arxiv.org/pdf/2211.12204v2.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":"online-size-ramsey-numbers-path-vs-c-4","repo_url":"https://github.com/urojony/c4pn","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}