{"paper":{"title":"On the maximum size of $(a,b)$-town (mod $k$) families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hanlin Zou","submitted_at":"2026-06-02T13:13:49Z","abstract_excerpt":"For integers $n \\geq k \\geq 2$ and $0 \\leq a,b \\leq k-1$, let $m_{k,n}(a,b)$ denote the maximum size of an $(a,b)$-town (mod $k$) family of an $n$-element set, a collection of subsets of whose cardinalities are congruent to $a$ modulo $k$ and whose pairwise intersections are congruent to $b$ modulo $k$. This notion generalizes the classical Oddtown and Eventown problems.\n  We prove that $m_{k,n}(a,b)\\leq n$ whenever $a\\not\\equiv b\\pmod{k}$, thereby resolving a conjecture of Veselinov and Marinov. We also disprove another conjecture of theirs by showing that $m_{3,11}(2,2)>m_{3,11}(1,1)$. For t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.03613","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/2606.03613/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"}