{"paper":{"title":"A short note on supersaturation for oddtown and eventown","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jason O'Neill","submitted_at":"2021-09-21T02:57:32Z","abstract_excerpt":"Given a collection $\\mathcal{A}$ of subsets of an $n$ element set, let $\\text{op}(\\mathcal{A})$ denote the number of distinct pairs $A,B \\in \\mathcal{A}$ for which $|A \\cap B|$ is odd. For $s \\in \\{1,2\\}$, we prove $\\text{op}(\\mathcal{A}) \\geq s \\cdot 2^{\\lfloor n/2 \\rfloor-1}$ for any collection $\\mathcal{A}$ of $2^{\\lfloor n/2 \\rfloor}+s$ even-sized subsets of an $n$ element set. We also prove $\\text{op}(\\mathcal{A}) \\geq 3$ for any collection $\\mathcal{A}$ of $n+1$ odd-sized subsets of an $n$ element set that. Moreover, we show that both of these results are best possible. We then consider "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.09925","kind":"arxiv","version":2},"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/2109.09925/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"}