{"paper":{"title":"On a conjecture regarding the product version of the Hilton-Milner theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lihua Feng, Lu Lu, Qifan Wang, Xucheng Bu, Zejun Huang","submitted_at":"2026-07-07T16:10:29Z","abstract_excerpt":"Recently, Frankl and Wang considered a product version of the classical Hilton-Milner theorem. They conjectured that, if $\\mathcal{F} \\subset \\binom{[n]}{k}$ and $\\mathcal{G} \\subset \\binom{[n]}{\\ell}$ are non-trivial cross-intersecting families with $n \\geq 2k > 2\\ell \\geq 4$, the maximum of $|\\mathcal{F}||\\mathcal{G}|$ is attained by the natural Hilton-Milner-type configurations.\n  In this paper, we present two main results concerning this conjecture. Firstly, we show that the conjecture does not hold in general. By introducing a two-center construction, we prove that for every fixed integer"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06443","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/2607.06443/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"}