{"paper":{"title":"A linear upper bound on the $\\mathbb{Z}_p$-Ramsey number of graphs with sufficiently large $2$-packing","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Simmons, Emily Heath","submitted_at":"2026-05-20T23:38:08Z","abstract_excerpt":"Given a positive integer $k$ and graph $G$, the $\\mathbb{Z}_k$-Ramsey number $R(G,\\mathbb{Z}_k)$ is the least $N$ (if it exists) such that every coloring $f:E(K_N)\\rightarrow \\mathbb{Z}_k$ contains a copy $G'$ of $G$ such that $\\sum_{e\\in E(G')}f(e)=0$. Motivated by a question of Caro and Mifsud, we study the $\\mathbb{Z}_k$-Ramsey number of graphs with a sufficiently large 2-packing, i.e. a set of vertices $S\\subseteq V(G)$ such that $N[u]\\cap N[v]=\\emptyset$ for all distinct $u,v\\in S$. In particular, we prove that $R(G,\\mathbb{Z}_p)\\leq n+6p-9$ for all $n$-vertex graphs $G$ and all primes $p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21817","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.21817/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}