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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.21817","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T23:38:08Z","cross_cats_sorted":[],"title_canon_sha256":"514e3ed1d37b478fdd989908d640fd0a9591454f9710be7b11bc11d34170d825","abstract_canon_sha256":"1aac11ded0c05237123afa382f9bb7ac38a99cf0d31865501ed69d94fe84cc95"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:04:09.183857Z","signature_b64":"1cgtnE1bWzNI+ZfomTk6CImvVSXTfDTqQuhIsHzLpBAs4sGi0GS02zzCM0oIrxY+b/yzQKD1wIRePut2F+NOCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f1374d46810c33bee8b9a2758bf4e2ed19aee8c7e1eac75226e46ff5078e312","last_reissued_at":"2026-05-22T01:04:09.182823Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:04:09.182823Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.21817","created_at":"2026-05-22T01:04:09.182960+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.21817v1","created_at":"2026-05-22T01:04:09.182960+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.21817","created_at":"2026-05-22T01:04:09.182960+00:00"},{"alias_kind":"pith_short_12","alias_value":"N4JXJVDICDBT","created_at":"2026-05-22T01:04:09.182960+00:00"},{"alias_kind":"pith_short_16","alias_value":"N4JXJVDICDBTX3UL","created_at":"2026-05-22T01:04:09.182960+00:00"},{"alias_kind":"pith_short_8","alias_value":"N4JXJVDI","created_at":"2026-05-22T01:04:09.182960+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/N4JXJVDICDBTX3ULTITVRP2OF3","json":"https://pith.science/pith/N4JXJVDICDBTX3ULTITVRP2OF3.json","graph_json":"https://pith.science/api/pith-number/N4JXJVDICDBTX3ULTITVRP2OF3/graph.json","events_json":"https://pith.science/api/pith-number/N4JXJVDICDBTX3ULTITVRP2OF3/events.json","paper":"https://pith.science/paper/N4JXJVDI"},"agent_actions":{"view_html":"https://pith.science/pith/N4JXJVDICDBTX3ULTITVRP2OF3","download_json":"https://pith.science/pith/N4JXJVDICDBTX3ULTITVRP2OF3.json","view_paper":"https://pith.science/paper/N4JXJVDI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.21817&json=true","fetch_graph":"https://pith.science/api/pith-number/N4JXJVDICDBTX3ULTITVRP2OF3/graph.json","fetch_events":"https://pith.science/api/pith-number/N4JXJVDICDBTX3ULTITVRP2OF3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N4JXJVDICDBTX3ULTITVRP2OF3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N4JXJVDICDBTX3ULTITVRP2OF3/action/storage_attestation","attest_author":"https://pith.science/pith/N4JXJVDICDBTX3ULTITVRP2OF3/action/author_attestation","sign_citation":"https://pith.science/pith/N4JXJVDICDBTX3ULTITVRP2OF3/action/citation_signature","submit_replication":"https://pith.science/pith/N4JXJVDICDBTX3ULTITVRP2OF3/action/replication_record"}},"created_at":"2026-05-22T01:04:09.182960+00:00","updated_at":"2026-05-22T01:04:09.182960+00:00"}