{"paper":{"title":"Balanced Sperner families via the topological Tverberg theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chong Shangguan, Yulin Yang, Zixiang Xu","submitted_at":"2026-06-09T13:59:23Z","abstract_excerpt":"For every prime power \\(r\\ge 2\\), we show that any Sperner family \\(\\mathcal F\\subseteq 2^{[n]}\\) with \\(|\\mathcal F|\\ge (r-1)n+1\\) contains \\(r\\) pairwise disjoint nonempty subfamilies whose unions are all equal and whose intersections are all equal. For \\(r=2\\), this confirms a conjecture of Heged\\\"{u}s, with the sharp threshold \\(n+1\\). In this purely combinatorial problem, our proof combines a multilinear polynomial method, a continuity argument, and the topological Tverberg theorem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.10885","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/2606.10885/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"}