{"paper":{"title":"Martin's axiom and $\\omega_1^2 \\longrightarrow (\\omega_1^2, 3)^2$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Mohammad Golshani","submitted_at":"2026-08-13T13:16:23Z","abstract_excerpt":"Starting with CH and Hajnal's coloring, we show that a standard finite-support iteration of $\\sigma$-centered forcing notions gives a model of $\n  MA_{\\omega_1}(\\sigma$-centered$)+2^{\\aleph_0}=\\aleph_2\n  +\\omega_1^2\\nrightarrow(\\omega_1^2,3)^2.$ We also isolate a simple forcing-preservation principle: the same ground-model coloring remains a witness after forcing with any poset whose subfamilies of size at most $\\omega_1$ are countable unions of linked sets. Under $\\text{MA}_{\\omega_1}$, every c.c.c. forcing has this local property, so every existing witness is preserved by every c.c.c. forcin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.13213","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.13213/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":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}