{"paper":{"title":"Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Tur\\'{a}n Methods","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Heng Li, Xizhi Liu","submitted_at":"2026-05-23T10:40:42Z","abstract_excerpt":"We revisit a conjecture of Mubayi that was proposed as a hypergraph analogue of the Li--Li algebraic proof of Tur\\'{a}n's theorem. The conjecture compares a polynomial ideal generated by multipartite 3-graphs with a differentiated diagonal-vanishing ideal. We show that the proposed equality fails for every non-vacuous choice of parameters. The obstruction is structural: diagonal vanishing does not remember the missing codegree-star condition that drives Mubayi's hypergraph problem.\n  We then give a replacement in edge-variable rings using monomial cover ideals. For ordinary forbidden-family Tu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24507","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/2605.24507/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"}