{"paper":{"title":"An Improved Tur\\'an Exponent for 2-Complexes","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Maya Sankar","submitted_at":"2024-08-16T21:54:29Z","abstract_excerpt":"The topological Tur\\'an number $\\mathrm{ex}_{\\hom}(n,X)$ of a 2-dimensional simplicial complex $X$ asks for the maximum number of edges in an $n$-vertex 3-uniform hypergraph containing no triangulation of $X$ as a subgraph. We prove that the Tur\\'an exponent of any such space $X$ is at most $8/3$, i.e., that $\\mathrm{ex}_{\\hom}(n,X)\\leq Cn^{8/3}$ for some constant $C=C(X)$. This improves on the previous exponent of $3-1/5$, due to Keevash, Long, Narayanan, and Scott. Additionally, we present new streamlined proofs of the asymptotically tight upper bounds for the topological Tur\\'an numbers of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.09029","kind":"arxiv","version":2},"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"}