{"paper":{"title":"Cutsets in ${\\mathcal P}(X)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.CO","authors_text":"Bill Sands, John Ginsburg","submitted_at":"2025-08-13T22:15:28Z","abstract_excerpt":"For any set $X$, ${\\mathcal P}(X)$ denotes the collection of all subsets of $X$, ordered by inclusion. A {\\it cutset} in ${\\mathcal P}(X)$ is a subset of ${\\mathcal P}(X)$ which meets every maximal chain of ${\\mathcal P}(X)$. A cutset is non-trivial if it does not contain $X$ or the empty set. Our main result is the following.\n  Theorem 1: Let $X$ be an infinite set of cardinality $\\kappa$. Every non-trivial cutset in ${\\mathcal P}(X)$ contains a chain of cardinality $\\kappa^+$ and an antichain of cardinality $2^{\\kappa}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.10221","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/2508.10221/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"}