{"paper":{"title":"A Spectral Hilton--Milner--Frankl Theorem for $t$-Intersecting Families","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lihua Deng, Lu Lu, Rongrong Lu, Xucheng Bu","submitted_at":"2026-08-07T05:03:22Z","abstract_excerpt":"Keevash, Lenz, and Mubayi proved a spectral Erd\\H{o}s--Ko--Rado theorem, showing that, for sufficiently large $n$, the complete $t$-star uniquely maximizes the adjacency-tensor spectral radius among all $t$-intersecting $k$-uniform families.\n  In this paper, we establish a spectral Hilton--Milner--Frankl theorem for nontrivial $t$-intersecting families in the explicit range $1\\le t\\le k-2$ and $n\\ge 100\\cdot 2^k k^7$. More precisely, we prove that, for every nontrivial $t$-intersecting $k$-uniform family $\\mathcal F$, the spectral radius satisfies \\[\n  \\rho(\\mathcal F)\\le\n  \\max\\{\\rho(\\mathcal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06810","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/2608.06810/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"}