{"paper":{"title":"Measurable Brooks's Theorem for Directed Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Borel directed graphs of maximum degree d admit measurable d-dicolorings unless they contain the complete symmetric digraph on d+1 vertices.","cross_cats":["math.CO"],"primary_cat":"math.LO","authors_text":"Cecelia Higgins","submitted_at":"2024-05-02T04:08:45Z","abstract_excerpt":"We prove a descriptive version of Brooks's theorem for directed graphs. In particular, we show that, if $D$ is a Borel directed graph on a standard Borel space $X$ such that the maximum degree of each vertex is at most $d \\geq 3$, then unless $D$ contains the complete symmetric directed graph on $d + 1$ vertices, $D$ admits a $\\mu$-measurable $d$-dicoloring with respect to any Borel probability measure $\\mu$ on $X$, and $D$ admits a $\\tau$-Baire-measurable $d$-dicoloring with respect to any Polish topology $\\tau$ compatible with the Borel structure on $X$. We also prove a definable version of "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"if D is a Borel directed graph on a standard Borel space X such that the maximum degree of each vertex is at most d ≥ 3, then unless D contains the complete symmetric directed graph on d + 1 vertices, D admits a μ-measurable d-dicoloring with respect to any Borel probability measure μ on X, and D admits a τ-Baire-measurable d-dicoloring with respect to any Polish topology τ compatible with the Borel structure on X.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The directed graph D must be Borel as a subset of X × X on a standard Borel space X; this Borel assumption is what permits the existence of measurable colorings with respect to arbitrary Borel measures and compatible Polish topologies (abstract, first sentence of main result).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Borel directed graphs of bounded degree admit mu-measurable and Baire-measurable d-dicolorings unless containing the complete symmetric digraph on d+1 vertices, plus a definable Gallai theorem for list dicolorings.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Borel directed graphs of maximum degree d admit measurable d-dicolorings unless they contain the complete symmetric digraph on d+1 vertices.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ac106f239121593fd3f5ac2fb6258c05c8fe757e0f90d7601daeb9b65d452d43"},"source":{"id":"2405.00991","kind":"arxiv","version":4},"verdict":{"id":"e2b05219-09ba-4d32-b576-277c9bc26e6d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-24T01:36:17.725709Z","strongest_claim":"if D is a Borel directed graph on a standard Borel space X such that the maximum degree of each vertex is at most d ≥ 3, then unless D contains the complete symmetric directed graph on d + 1 vertices, D admits a μ-measurable d-dicoloring with respect to any Borel probability measure μ on X, and D admits a τ-Baire-measurable d-dicoloring with respect to any Polish topology τ compatible with the Borel structure on X.","one_line_summary":"Borel directed graphs of bounded degree admit mu-measurable and Baire-measurable d-dicolorings unless containing the complete symmetric digraph on d+1 vertices, plus a definable Gallai theorem for list dicolorings.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The directed graph D must be Borel as a subset of X × X on a standard Borel space X; this Borel assumption is what permits the existence of measurable colorings with respect to arbitrary Borel measures and compatible Polish topologies (abstract, first sentence of main result).","pith_extraction_headline":"Borel directed graphs of maximum degree d admit measurable d-dicolorings unless they contain the complete symmetric digraph on d+1 vertices."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2405.00991/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":2,"sample":[{"doi":"","year":2024,"title":"[BCG+24] S. Brandt, Y. J. Chang, J. Greb´ ık, C. Grunau, V. Rozhoˇ n, and Z. Vidny´ anszky,On homomor- phism graphs, Forum Math. Pi12(2024), e10. [Ber19] A. Bernshteyn,Measurable versions of the Lov´ ","work_id":"24450fc0-5d61-4810-9fec-ea17b079141d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1999,"title":"Borel order dimension","work_id":"64fe3eaa-f4f6-4a08-9973-129d704039ff","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":2,"snapshot_sha256":"4b85a9c1c0a794b812e27db7d0b2134bb704d0a63e56fecf78a97c937226c8c9","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"b4f05b2730c06bd7755d94d7bd8d17845268266095959fed49269c2fd4027e63"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}