{"paper":{"title":"Squares, ascent paths, and chain conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Chris Lambie-Hanson, Philipp L\\\"ucke","submitted_at":"2017-09-13T20:54:42Z","abstract_excerpt":"With the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todor\\v{c}evi\\'{c}'s principle $\\square(\\kappa)$ implies an indexed version of $\\square(\\kappa,\\lambda)$, we show that for all infinite, regular cardinals $\\lambda<\\kappa$, the principle $\\square(\\kappa)$ implies the existence of a $\\kappa$-Aronszajn tree containing a $\\lambda$-ascent path. We then provide a complete picture of the consistency strengths of sta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.04537","kind":"arxiv","version":1},"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"}