{"paper":{"title":"Combinatorial dichotomies and cardinal invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Dilip Raghavan, Stevo Todorcevic","submitted_at":"2013-05-24T16:33:20Z","abstract_excerpt":"Assuming the P-ideal dichotomy, we attempt to isolate those cardinal characteristics of the continuum that are correlated with two well-known consequences of the proper forcing axiom. We find a cardinal invariant $\\mathfrak{x}$ such that the statement that $\\mathfrak{x} > {\\omega}_{1}$ is equivalent to the statement that 1, $\\omega$, ${\\omega}_{1}$, $\\omega \\times {\\omega}_{1}$, and ${\\left[{\\omega}_{1}\\right]}^{< \\omega}$ are the only cofinal types of directed sets of size at most ${\\aleph}_{1}$. We investigate the corresponding problem for the partition relation ${\\omega}_{1} \\rightarrow ({\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.5783","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"}