{"paper":{"title":"Tur\\'{a}n results for posets and their alternating cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Geir Agnarsson, John B. Kent","submitted_at":"2026-06-23T17:54:28Z","abstract_excerpt":"For a partially ordered set ${\\mathbb{P}} = (X,\\leq)$ there exist hypergraphs where the vertices are the set of ordered tuples of either all incomparable elements of ${\\mathbb{P}}$ or all the critical pairs of ${\\mathbb{P}}$, and the edges are formed by the duals of either all the alternating cycles of ${\\mathbb{P}}$ or all the strict alternating cycles of ${\\mathbb{P}}$. The weak chromatic numbers of these hypergraphs are all equal to the order dimension of ${\\mathbb{P}}$. Here are established upper bounds on the number of strict alternating cycles a poset ${\\mathbb{P}}=(X,\\leq)$ can have in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24877","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/2606.24877/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"}