{"paper":{"title":"Erd\\H{o}s--Ko--Rado and Hilton--Milner Theorems in the Partition Lattice","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Haixiang Zhang, Jiaqi Liao, Mengyu Cao","submitted_at":"2026-08-06T12:21:18Z","abstract_excerpt":"Let $M_n=M(K_{n+1})$ be the graphic matroid of the complete graph, and let $\\mathcal{F}_k(M_n)$ be its rank-$k$ flats. We study families $\\mathcal{A}\\subseteq\\mathcal{F}_k(M_n)$ satisfying $\\mathrm{rk}(A\\wedge B)\\ge t$ for all $A,B\\in\\mathcal{A}$. For $t=1$, this problem is exactly equivalent to Czabarka's partition-EKR conjecture, first introduced in print by P.~L. Erd\\H{o}s and L.~A. Sz\\'ekely~\\cite{ErdosSzekelyHigher}. We prove the corresponding Erd\\H{o}s--Ko--Rado theorem in the explicit linear range $n+1\\ge8k$, giving a constant-factor advance toward the conjectured sharp range $n\\ge2k$. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.05951","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/2608.05951/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"}