{"paper":{"title":"Recursive upper bounds for the vertex online Ramsey game with applications to hypergraph Ramsey numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"D\\'aniel Dob\\'ak, Eion Mulrenin","submitted_at":"2026-05-15T20:16:13Z","abstract_excerpt":"The classical recursive upper bound on hypergraph Ramsey numbers due to Erd\\H{o}s and Rado states that for $2 \\leq k < s \\leq t$,\n  \\[\n  r_k(s,t) \\leq 2^{\\binom{r_{k-1}(s-1,t-1)}{k-1}}.\n  \\]\n  In 2010, Conlon, Fox, and Sudakov introduced the so-called vertex online Ramsey numbers $\\tilde{r}(s,t)$ for graphs to obtain a quantitative improvement over this bound when $k=3$.\n  In this note, we show that the natural hypergraph generalization $\\tilde{r}_k(s,t)$ of the vertex online Ramsey numbers satisfy an improved recurrence\n  \\[\n  \\tilde{r}_k(s,t) \\leq 2^{(1+o(1))\\tilde{r}_{k-1}(s-1,t-1)}.\n  \\]\n "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.16607","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/2605.16607/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-19T19:21:56.800109Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.596866Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"a46fd52921679d6bf2f8b1667b31c6cf65e959f8391aff6d8ed8d66ae1cacaf1"},"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"}