{"paper":{"title":"2-complexes with large 2-girth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.AT","authors_text":"Dominic Dotterrer, Larry Guth, Matthew Kahle","submitted_at":"2015-09-13T17:06:03Z","abstract_excerpt":"The 2-girth of a 2-dimensional simplicial complex $X$ is the minimum size of a non-zero 2-cycle in $H_2(X, \\mathbb{Z}/2)$. We consider the maximum possible girth of a complex with $n$ vertices and $m$ 2-faces. If $m = n^{2 + \\alpha}$ for $\\alpha < 1/2$, then we show that the 2-girth is at most $4 n^{2 - 2 \\alpha}$ and we prove the existence of complexes with 2-girth at least $c_{\\alpha, \\epsilon} n^{2 - 2 \\alpha - \\epsilon}$. On the other hand, if $\\alpha > 1/2$, the 2-girth is at most $C_{\\alpha}$. So there is a phase transition as $\\alpha$ passes 1/2.\n  Our results depend on a new upper boun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03871","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"}