{"paper":{"title":"Minimum degree in simplicial complexes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bjarne Sch\\\"ulke, Christian Reiher","submitted_at":"2025-01-02T15:21:08Z","abstract_excerpt":"Given $d\\in\\mathbb{N}$, let $\\alpha(d)$ be the largest real number such that every abstract simplicial complex $\\mathcal{S}$ with $0<\\vert\\mathcal{S}\\vert\\leq\\alpha(d)\\vert V(\\mathcal{S})\\vert$ has a vertex of degree at most $d$. We extend previous results by Frankl, Frankl and Watanabe, and Piga and Sch\\\"ulke by proving that for all integers $d$ and $m$ with $d\\geq m\\geq 1$, we have $\\alpha(2^d-m)=\\frac{2^{d+1}-m}{d+1}$. Similar results were obtained independently by Li, Ma, and Rong."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.01294","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/2501.01294/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"}