{"paper":{"title":"Sharp Bounds on the Independence Number of Simplicial Spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ethan X. Fang, Hailun Zheng, Jes\\'us A. De Loera, Junwei Lu, Tingzhou Wei","submitted_at":"2026-08-06T03:40:06Z","abstract_excerpt":"We study the maximum size of an independent set in the graph of a simplicial sphere. Let $\\beta(d,n)$ denote this maximum over all simplicial $(d-1)$-spheres on $n$ vertices, and let $\\alpha(d,n)$ denote the maximum restricted to flag $(d-1)$-spheres. For every fixed $d\\geq4$, we prove $\\beta(d,n)=n-\\Theta(n^{1/\\lfloor d/2\\rfloor})$. For flag spheres, we show $\\alpha(d,n)\\geq n-4\\sqrt n+O(1)$ for all $d\\geq4$ and determine the correct asymptotic order $\\alpha(d,n)=n-\\Theta(\\sqrt n)$ for dimensions $d=4,5$. We also investigate the independence sets of Bier spheres and show that, in contrast to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.05568","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/2608.05568/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"}