Kalai’s tight-tree extremal bound
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The question is whether 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.
Why it matters and what counts as progress
Why it matters: It predicts the sharp density threshold for forcing every tight uniform hypergraph tree, including a stronger shadow-based bound.
What counts as progress: Prove the sharp edge or shadow bound for every uniformity and tight tree, with constructions showing the coefficient is optimal.
How the problem developed
Gil Kalai Proposed the extremal bound for r-uniform hypergraphs avoiding a tight r-tree; for r=2 it specializes to the Erdős–Sós conjecture.
- 7 Sept 2026 · Theorem [2]
Dhruv Mubayi and Jacques Verstraëte Proved the stronger shadow bound |E(H)|≤(t−1)|∂H|/r for a T-free r-graph and tight r-tree T with t edges; the paper credits GPT-6 Astra with discovering the proof.
AI contribution
Kalai's Conjecture for Tight Trees
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.
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