{"paper":{"title":"Random partition for Tokushige's $r$-wise intersecting conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Lihua Feng, Yongjiang Wu","submitted_at":"2026-06-30T03:04:41Z","abstract_excerpt":"Let $r\\ge 3$ and let $1>p_1\\ge p_2\\ge\\cdots\\ge p_n>0$. Let $\\mu_{\\mathbf p}$ denote the product measure on $2^{[n]}$ where each coordinate $i$ is included independently with probability $p_i$. A family $\\mathcal A\\subseteq 2^{[n]}$ is $r$-wise intersecting if $A_1\\cap\\cdots\\cap A_r\\neq\\emptyset$ for all $A_1,\\ldots,A_r\\in\\mathcal A$. In 2022, Tokushige proved that if $p_2<\\frac{r-1}{r}$, then every $r$-wise intersecting family $\\mathcal{A}\\subseteq 2^{[n]}$ satisfies $\\mu_{\\mathbf p}(\\mathcal{A})\\le p_1$, with equality only for stars centred at coordinates of maximum probability. He conjecture"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31075","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/2606.31075/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"}