{"paper":{"title":"Some new Bollob\\'as-type inequalities","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Erfei Yue","submitted_at":"2024-05-27T20:20:29Z","abstract_excerpt":"A family of disjoint pairs of finite sets $\\mathcal{P}=\\{(A_i,B_i)\\mid i\\in[m]\\}$ is called a Bollob\\'as system if $A_i\\cap B_j\\neq\\emptyset$ for every $i\\neq j$, and a skew Bollob\\'as system if $A_i\\cap B_j\\neq\\emptyset$ for every $i<j$. Bollob\\'as proved that for a Bollob\\'as system, the inequality \\begin{equation*} \\sum_{i=1}^m\\binom{|A_i|+|B_i|}{|A_i|}^{-1}\\leq 1 \\end{equation*} holds. Heged\\\"{u}s and Frankl generalized this theorem to skew Bollob\\'as systems with the inequality \\begin{equation*} \\sum_{i=1}^m\\binom{|A_i|+|B_i|}{|A_i|}^{-1}\\leq 1+n, \\end{equation*} provided $A_i,B_i\\subsete"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.17639","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/2405.17639/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"}