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The classical oddtown and eventown problems study the cases $\\boldsymbol{\\alpha} = (1, 0)$ and $(0, 0)$, respectively. We determine the sharp asymptotics of $f_{\\boldsymbol{\\alpha}}(n)$ for all $\\boldsymbol{\\alpha}$, answering questions of Johnston--O'Neill and Wei--Zhang--Ge.\n  We also study a symmetric variant $g_{\\"},"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":"2606.11139","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-09T17:29:46Z","cross_cats_sorted":[],"title_canon_sha256":"6e3ea76135d6813697981c0897b37bcef6a16021fcc58841a57933b449338ebf","abstract_canon_sha256":"9d9c21d5d38a884e2a4cbf7d2989d7e6167f43ec2794ddb8b0bdeb3106e917ea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:11:13.164449Z","signature_b64":"noffvODaoxzC6o76igzNmeOI3ugC8AnNApEh3SiQR2VTz+z7Q0YXBVyNCDC+xinsG1ch492EmWDmKMwq0CRYCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d64733a4b6b0225301825e0fb2a994fbf011bb6d7908391724baf35f06cdfdf","last_reissued_at":"2026-06-10T01:11:13.163814Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:11:13.163814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp bounds on $k$-wise generalizations of oddtowns and eventowns","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lan Wei, Minghui Ouyang, Zichao Dong","submitted_at":"2026-06-09T17:29:46Z","abstract_excerpt":"For $\\boldsymbol{\\alpha} = (\\alpha_1, \\dots, \\alpha_k) \\in \\mathbb{F}_2^k$, an $\\boldsymbol{\\alpha}$-town is a set family in which every $i$-wise intersection has parity $\\alpha_i$. 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