{"paper":{"title":"Hindman and Owings-like theorems without the Axiom of Choice","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The uncountable analog of Hindman's theorem fails for the additive group of the reals under ZF and for Q-vector spaces of uncountable dimension under DC when the dimension is not well-orderable.","cross_cats":["math.CO"],"primary_cat":"math.LO","authors_text":"David J. Fern\\'andez Bret\\'on, Eliseo Sarmiento Rosales, Jos\\'e A. Guzm\\'an-Vega","submitted_at":"2026-03-28T06:59:02Z","abstract_excerpt":"We investigate Hindman- and Owings-type Ramsey-theoretic statements in Zermelo-Fraenkel set theory without the Axiom of Choice, with some occasional extra assumptions (such as the Axiom of Dependent Choice and/or the Axiom of Determinacy). We study several variations of Hindman's theorem on $\\mathbb Q$-vector spaces; notably, we show that the uncountable analog of Hindman's theorem fails for the additive group of $\\mathbb R$ (under ZF), and for $\\mathbb Q$-vector spaces of uncountable dimension (under DC if such dimension is not well-orderable), among other results. In contrast, for Owings-typ"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"the uncountable analog of Hindman's theorem fails for the additive group of R (under ZF), and for Q-vector spaces of uncountable dimension (under DC if such dimension is not well-orderable)","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the uncountable dimension is not well-orderable when working under DC; if every set of reals is well-orderable then the negative result may not hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Uncountable Hindman theorems fail in ZF and under DC for non-well-orderable dimensions, while Owings configurations hold under AD.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The uncountable analog of Hindman's theorem fails for the additive group of the reals under ZF and for Q-vector spaces of uncountable dimension under DC when the dimension is not well-orderable.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"38c0f26d8aa6ed24ec9edca5b58bd0132429f416c51af5cd70e7527e29d21055"},"source":{"id":"2603.27163","kind":"arxiv","version":4},"verdict":{"id":"fb368cbc-fed9-4589-a9b5-5711d8301dd3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T22:19:22.789226Z","strongest_claim":"the uncountable analog of Hindman's theorem fails for the additive group of R (under ZF), and for Q-vector spaces of uncountable dimension (under DC if such dimension is not well-orderable)","one_line_summary":"Uncountable Hindman theorems fail in ZF and under DC for non-well-orderable dimensions, while Owings configurations hold under AD.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the uncountable dimension is not well-orderable when working under DC; if every set of reals is well-orderable then the negative result may not hold.","pith_extraction_headline":"The uncountable analog of Hindman's theorem fails for the additive group of the reals under ZF and for Q-vector spaces of uncountable dimension under DC when the dimension is not well-orderable."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.27163/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":3,"snapshot_sha256":"46a0b7f6421d2a81ab66e2907e3f7c56361eb6466859ab992c51549a707503ac"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}