{"paper":{"title":"On model-theoretic tree properties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.LO","authors_text":"Artem Chernikov, Nicholas Ramsey","submitted_at":"2015-05-03T17:14:53Z","abstract_excerpt":"We study model theoretic tree properties ($\\text{TP}, \\text{TP}_1, \\text{TP}_2$) and their associated cardinal invariants ($\\kappa_{\\text{cdt}}, \\kappa_{\\text{sct}}, \\kappa_{\\text{inp}}$, respectively). In particular, we obtain a quantitative refinement of Shelah's theorem ($\\text{TP} \\Rightarrow \\text{TP}_1 \\lor \\text{TP}_2$) for countable theories, show that $\\text{TP}_1$ is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak $k-\\text{TP}_1$ is equivalent to $\\text{TP}_1$ (answering a question of Kim and Kim). Besides, we give a charact"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.00454","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":""},"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"}