{"paper":{"title":"A gap theorem for non-trivial maximal intersecting families and an exact weighted asymptotic","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"El'mira Yu. Kalimulina","submitted_at":"2026-07-17T15:15:13Z","abstract_excerpt":"Let $D_n$ be the disjointness graph on the nonempty subsets of $[n]$, whose independent sets are exactly the intersecting families on $[n]$. We study the weighted independent-set polynomial $W(n)=\\sum_F\\prod_{S\\in F}w(S)$, the sum running over these families, for the doubly exponential weight $w(S)=2^{2^{n-|S|}}-1$. The kernel-bearing (trivial) part $Z_\\cap(n)$ is exact by inclusion-exclusion and satisfies $Z_\\cap(n)\\sim n\\cdot 2^{3^{n-1}}$. For the kernel-free remainder we prove the exact prefactor $R(n)=(3/4+o(1))n\\cdot 2^{3^{n-1}-2^{n-1}+2}$, whence $\\log_2(Z_\\cap(n)/R(n))=2^{n-1}-2+\\log_2("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16040","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/2607.16040/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"}