What was already known
Kalai’s bound was known for selected tree classes, including stars and trees with at most four edges, and for r-partite hosts. Earlier tight-path bounds were weaker.
AI involvement
Mubayi and Verstraëte prompted GPT-6 Astra first for tight paths and then for all tight trees. The model found the main proof; the authors checked it, rewrote substantial text and added explanations.
The model found the main proof after human prompts specified the path extension and full hypergraph target.
- Supporting source (arxiv.org) — v1 submission history through 18 October 2026
Importance
Proves Kalai’s conjectured edge bound for all uniformities and tight trees, including a stronger shadow form.
Limitations
The authors checked the main proof but explicitly did not verify the model’s proposed equality characterization or stronger counting claims; those are outside this assessment.
Technical verification
Both authors explicitly report checking the main proof. Equality and stronger counting claims they did not verify are excluded.
Independent checking
author check of full proof, rewrite and added explanations. Theorem 2.1 tight-tree shadow bound
- Checker
- Dhruv Mubayi and Jacques Verstraëte
- Scope
- For n≥r≥2, t≥1 and any n-vertex r-uniform hypergraph H containing no tight r-tree T with t edges, |E(H)|≤(t−1)|∂H|/r≤(t−1)binom(n,r−1)/r, where ∂H is the (r−1)-shadow. The bound is attained for infinitely many n.
- Basis
- Reported by the linked checker
Human understanding
The authors explain path rotation, rooted states, leaf addition and induction through the full shadow bound.
Correction and publication history
The checked tight-tree shadow bound excludes the proposed equality and counting claims.
Read this published revision