{"ID":6023533,"CreatedAt":"2026-07-08T01:00:23.257252134Z","UpdatedAt":"2026-07-10T11:42:49.717029521Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.06171","arxiv_id":"2607.06171","title":"The singleton hypergraph is extremal for the Isolation Lemma","abstract":"Let $H$ be an inclusion-free hypergraph on $n$ vertices. A weight assignment $w:[n]\\to[d]$ is isolating if there is a unique edge $e$ whose weight $w(e) = \\sum_{i \\in e} w(i)$ is minimum. We show that the number of isolating weight assignments is at least $$ n\\sum_{j=0}^{d-1} j^{n-1}, $$ a bound which is attained with equality by the hypergraph consisting of the $n$ singleton edges. This proves the conjecture stated in Faber \u0026 Harris (2018). We also prove the bound for a more general class of edge-weight objectives, including arbitrary edge offsets.","short_abstract":"Let $H$ be an inclusion-free hypergraph on $n$ vertices. A weight assignment $w:[n]\\to[d]$ is isolating if there is a unique edge $e$ whose weight $w(e) = \\sum_{i \\in e} w(i)$ is minimum. We show that the number of isolating weight assignments is at least $$ n\\sum_{j=0}^{d-1} j^{n-1}, $$ a bound which is attained with...","url_abs":"https://arxiv.org/abs/2607.06171","url_pdf":"https://arxiv.org/pdf/2607.06171v1","authors":"[\"Vance Faber\",\"David G. Harris\"]","published":"2026-07-07T11:49:20Z","proceeding":"math.CO","tasks":"[\"math.CO\",\"cs.DS\"]","methods":"[]","has_code":false}