{"paper":{"title":"On the Complexity of Embeddable Simplicial Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anna Gundert","submitted_at":"2018-12-20T09:48:17Z","abstract_excerpt":"This thesis addresses the question of the maximal number of $d$-simplices for a simplicial complex which is embeddable into $\\mathbb{R}^r$ for some $d \\leq r \\leq 2d$.\n  A lower bound of $f_d(C_{r + 1}(n)) = \\Omega(n^{\\lceil\\frac{r}{2}\\rceil})$, which might even be sharp, is given by the cyclic polytopes. To find an upper bound for the case $r=2d$ we look for forbidden subcomplexes. A generalization of the theorem of van Kampen and Flores yields those. Then the problem can be tackled with the methods of extremal hypergraph theory, which gives an upper bound of $O(n^{d+1-\\frac{1}{3^d}})$.\n  We "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.08447","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":""},"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"}