{"paper":{"title":"Chromatic number is not tournament-local","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ant\\'onio Gir\\~ao, Florian Lehner, Freddie Illingworth, Kevin Hendrey, Lukas Michel, Michael Savery, Raphael Steiner","submitted_at":"2023-05-24T21:41:18Z","abstract_excerpt":"Scott and Seymour conjectured the existence of a function $f \\colon \\mathbb{N} \\to \\mathbb{N}$ such that, for every graph $G$ and tournament $T$ on the same vertex set, $\\chi(G) \\geqslant f(k)$ implies that $\\chi(G[N_T^+(v)]) \\geqslant k$ for some vertex $v$. In this note we disprove this conjecture even if $v$ is replaced by a vertex set of size $\\mathcal{O}(\\log{\\lvert V(G)\\rvert})$. As a consequence, we answer in the negative a question of Harutyunyan, Le, Thomass\\'{e}, and Wu concerning the corresponding statement where the graph $G$ is replaced by another tournament, and disprove a relate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.15585","kind":"arxiv","version":2},"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/2305.15585/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"}