🤖 AI Summary
This work investigates the lower bound on the number of isolating weight assignments that render the minimum-weight edge unique in inclusion-free hypergraphs. Through combinatorial and hypergraph-theoretic analysis, it establishes that this lower bound is \( n\sum_{j=0}^{d-1} j^{\,n-1} \), and demonstrates that hypergraphs consisting solely of singleton edges precisely attain this bound. The result not only confirms a conjecture by Faber and Harris (2018) but also extends the extremal case of the Isolation Lemma to general objective functions with arbitrary edge offsets, thereby revealing the fundamental influence of hypergraph structure on the complexity of isolating weight assignments.
📝 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 & Harris (2018).
We also prove the bound for a more general class of edge-weight objectives, including arbitrary edge offsets.