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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1509.03871","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2015-09-13T17:06:03Z","cross_cats_sorted":["math.CO","math.PR"],"title_canon_sha256":"d4411113fe74d37a1cac221ed18c9c9534791f45b476aad4861ecb33e3939237","abstract_canon_sha256":"ee7d720525898408b1268f0a84e07a468c802257eefb43abdf5639b0b6e87e0b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:41.714338Z","signature_b64":"Le9IuHfJmjuKstRqsz5q6ZC5ix9iFOHbUTMIpvs+qE3QxK1+pV2K8aqLsaaHbis9WpuGW57ebA0w//oThYKTDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b23b74799dbc9ceb0328ef328ac64d9d99efe62affc6e8da5314810331d5218","last_reissued_at":"2026-05-18T00:40:41.713611Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:41.713611Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $H_2(X, \\mathbb{Z}/2)$. 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