{"paper":{"title":"Some questions on entangled linear orders","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Under the continuum hypothesis there exist n-entangled linear orders that are not n+1-entangled for every n.","cross_cats":["math.CO"],"primary_cat":"math.LO","authors_text":"Lorenzo Notaro, Maxwell Levine, Rapha\\\"el Carroy","submitted_at":"2025-07-23T13:38:59Z","abstract_excerpt":"Entangled linear orders were first introduced by Abraham and Shelah. Todor\\v{c}evi\\'c showed that these linear orders exist under $\\mathsf{CH}$. We prove the following results: (1) If $\\mathsf{CH}$ holds, then, for every $n > 0$, there is an $n$-entangled linear order which is not $(n+1)$-entangled. (2) If $\\mathsf{CH}$ holds, then there are two homeomorphic sets of reals $A, B \\subseteq \\mathbb{R}$ such that $A$ is entangled but $B$ is not $2$-entangled. (3) If $\\mathbb{R}\\subseteq \\mathrm{L}$, then there is an entangled $\\Pi_1^1$ set of reals. (4) If $\\diamondsuit$ holds, then there is a $2$"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"If CH holds, then for every n > 0 there is an n-entangled linear order which is not (n+1)-entangled; similar conditional existence statements hold for homeomorphic reals with differing entanglement, an entangled Pi1^1 set under V=L, and a 2-entangled non-separable order under diamond.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The background definitions of n-entangled linear order and of the relevant descriptive-set-theoretic complexity classes (Pi1^1) as introduced by Abraham-Shelah and used in the constructions, together with the assumption that the ambient universe satisfies the stated extra axioms (CH, V=L, diamond).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Under CH the authors construct n-entangled linear orders that fail to be (n+1)-entangled, plus homeomorphic reals with differing entanglement, an entangled Pi1^1 set under V=L, and a 2-entangled non-separable order under diamond.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Under the continuum hypothesis there exist n-entangled linear orders that are not n+1-entangled for every n.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"633f66cbb192716428c045997f2b3c6e7f3f171e44c365f50f268c23b5c66c40"},"source":{"id":"2507.17503","kind":"arxiv","version":2},"verdict":{"id":"35d5d630-64a7-44c7-9a8f-d8f439b65b09","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T03:37:14.892048Z","strongest_claim":"If CH holds, then for every n > 0 there is an n-entangled linear order which is not (n+1)-entangled; similar conditional existence statements hold for homeomorphic reals with differing entanglement, an entangled Pi1^1 set under V=L, and a 2-entangled non-separable order under diamond.","one_line_summary":"Under CH the authors construct n-entangled linear orders that fail to be (n+1)-entangled, plus homeomorphic reals with differing entanglement, an entangled Pi1^1 set under V=L, and a 2-entangled non-separable order under diamond.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The background definitions of n-entangled linear order and of the relevant descriptive-set-theoretic complexity classes (Pi1^1) as introduced by Abraham-Shelah and used in the constructions, together with the assumption that the ambient universe satisfies the stated extra axioms (CH, V=L, diamond).","pith_extraction_headline":"Under the continuum hypothesis there exist n-entangled linear orders that are not n+1-entangled for every n."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.17503/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":9,"sample":[{"doi":"","year":1985,"title":"On the consistency of some partition theorems for continuous colorings, and the structure of ℵ1-dense real order types","work_id":"ea34974d-f0c3-44e0-b8e7-3ce62b6f75be","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1973,"title":"All ℵ1-dense sets of reals can be isomorphic","work_id":"00ccda73-f333-4bea-adea-9604989c9769","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1940,"title":"Concerning similarity transformations of linearly ordered sets","work_id":"8584a407-ec4b-45b4-a931-fce69c475fc6","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1955,"title":"Order types and similarity transforma- tions","work_id":"d526ce59-3277-4f15-b8ac-b19952d47aa4","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Entangledness in Suslin lines and trees","work_id":"04be892b-ffd8-4c73-9e81-8a0398363167","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":9,"snapshot_sha256":"9c41a28054d3f53ec7c414a019a7c578aead42ebe3e4ddfeb1acb5e09c9448cf","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"ed4dd47a54823fdf28d4cbb4ca014ca430a059ccfcd6320c053f21fc2d947a02"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}