{"paper":{"title":"A Resolution of Erd\\H{o}s Problem 550 on Tree versus Complete Multipartite Ramsey Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eric Li (Trinity College, University of Cambridge)","submitted_at":"2026-06-22T17:45:05Z","abstract_excerpt":"We resolve Erd\\H{o}s Problem 550, originally asked as question (2) of Erd\\H{o}s, Faudree, Rousseau, and Schelp. Precisely, for fixed integers $k\\geq 2$ and $1\\leq m_1\\leq \\cdots \\leq m_k$, we prove that, for every sufficiently large $n$ and every $n$-vertex tree $T$, $R(T,K_{m_1,\\ldots,m_k}) \\leq (k-1)(R(T,K_{m_1,m_2})-1)+m_1$. The proof combines a new off-Tur\\'an tree-embedding theorem with a compactness-and-rounding theorem for represented bounded-rank hypergraph obstructions. The embedding theorem follows from Szemer\\'edi regularity and a local regular-matching embedding lemma of Hladk\\'y a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.23659","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/2606.23659/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"}