Sharp counts in the Isolation Lemma for inclusion-free hypergraphs
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How many weight assignments in [d]ⁿ isolate a unique minimum edge in a nonempty inclusion-free hypergraph, even when each edge has an arbitrary fixed real offset? The conjectured minimum is attained by the n singleton edges.
Why it matters and what counts as progress
Why it matters: The exact extremal count sharpens the probabilistic Isolation Lemma.
What counts as progress: The sharp lower count n∑ⱼ₌₀ᵈ⁻¹j^(n−1) for every such hypergraph and offset choice, or a hypergraph with fewer isolating assignments.
How the problem developed
- 24 Apr 2016 · Conjecture [1]
Vance Faber and David G. Harris Conjectured that the singleton-edge hypergraph minimizes the number of isolating assignments; proved the conjecture in several special cases and asymptotically.
- 7 Jul 2026 · Proof [2]
Vance Faber and David G. Harris, with ChatGPT assistance Proved the conjectured lower bound n∑ⱼ₌₀ᵈ⁻¹jⁿ⁻¹, attained by the n-singleton hypergraph, and extended it to arbitrary edge offsets.
AI contribution
The singleton hypergraph is extremal for the Isolation Lemma
For every nonempty inclusion-free hypergraph on n vertices, every d≥1 and arbitrary real edge offsets, Faber and Harris prove that at least n times the sum of j^(n−1) for j=0,…,d−1 assignments in [d]^n isolate a unique minimum edge. The n-singleton hypergraph attains equality.
Sources
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Publication dates and editorial changes.
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