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Now we show how the cited version of the Lov\\'asz Local Lemma applies to the space of random matchings in $K_{2n}$. We also prove tight upper bounds that asymptotically match the lower bound given by the Lov\\'asz Local Lemma. As a consequence, we give new proofs to results on the enumeration of $d$-regular graphs."},"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":"0905.3983","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-05-25T09:37:14Z","cross_cats_sorted":[],"title_canon_sha256":"e6e676db6ad700903f2b22b6cf741dc72a30af69af58632a916b4ee3f47f4921","abstract_canon_sha256":"f2741bb258bd7f16fe59e1c2df4469ac7e90c8988c9c52e24b1bd53e362822fb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:22.346135Z","signature_b64":"zc0ytIB+NPqzxRjo7p9Lb5fnDsVyfP2GkscKZFciClKxR3OgwDVn54lw2zTCZdcdqKx0kerm65dhl+Wx7zdKDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d07b4a713c5c1f9738db8deb4dc35426bd2c79c8c1022c9de60ed7bd1d84acce","last_reissued_at":"2026-05-18T02:58:22.345620Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:22.345620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A new asymptotic enumeration technique: the Lovasz Local Lemma","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Laszlo A. 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