{"paper":{"title":"A note on tilted Sperner families with patterns","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"D\\'aniel Gerbner, M\\'at\\'e Vizer","submitted_at":"2015-07-08T18:08:41Z","abstract_excerpt":"Let $p$ and $q$ be two nonnegative integers with $p+q>0$ and $n>0$. We call $\\mathcal{F} \\subset \\mathcal{P}([n])$ a \\textit{(p,q)-tilted Sperner family with patterns on [n]} if there are no distinct $F,G \\in \\mathcal{F}$ with: $$(i) \\ \\ p|F \\setminus G|=q|G \\setminus F|, \\ \\textrm{and}$$ $$(ii) \\ f > g \\ \\textrm{for all} \\ f \\in F \\setminus G \\ \\textrm{and} \\ g \\in G \\setminus F.$$\n  Long (\\cite{L}) proved that the cardinality of a (1,2)-tilted Sperner family with patterns on $[n]$ is $$O(e^{120\\sqrt{\\log n}}\\ \\frac{2^n}{\\sqrt{n}}).$$\n  We improve and generalize this result, and prove that th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02242","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":""},"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"}