{"paper":{"title":"Towards odd-sunflowers: temperate families and lightnings","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jan Petr, Pavel Turek","submitted_at":"2024-06-26T15:40:31Z","abstract_excerpt":"Motivated by odd-sunflowers, introduced recently by Frankl, Pach, and P{\\'a}lv{\\\"o}lgyi, we initiate the study of temperate families: a family $\\mathcal{F} \\subseteq \\mathcal{P}([n])$ is said to be \\emph{temperate} if each $A \\in \\mathcal{F}$ contains at most $|A|$ elements of $\\mathcal{F}$ as a proper subset.\n  We show that the maximum size of a temperate family is attained by the middle two layers of the hypercube $\\{0,1\\}^n$. As a more general result, we obtain that the middle $t+1$ layers of the hypercube maximise the size of a family $\\mathcal{F}$ such that each $A \\in \\mathcal{F}$ contai"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.18437","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/2406.18437/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"}