{"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 $\\alpha_i$. Denote by $f_{\\boldsymbol{\\alpha}}(n)$ the maximum size of an $\\boldsymbol{\\alpha}$-town on $[n]$. 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_{\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11139","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.11139/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}