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arXiv · 2609.08012

Kalai's Conjecture for Tight Trees

Dhruv Mubayi·Jacques Verstraete
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Abstract

Let $r \ge 2$ and $t \ge 1$. It is shown that if $T$ is an $r$-uniform tight tree with $t$ edges and $H$ is a $T$-free $r$-uniform hypergraph, then $|E(H)|\le (t-1)|\sh H|/r$, where $\sh H$ is the $(r-1)$-shadow of $H$. \iffalse Equality holds only for $(n,t + r - 2,r)$-designs.\fi The bound is tight infinitely often, and establishes Kalai's Conjecture, whose $r=2$ case is the Erd\H os-S\'os Conjecture. The proof was found by GPT-6 Astra, extending its method of proof for the Erd\H os-S\'os conjecture to the hypergraph setting. It is noteworthy that previous proofs of special cases of the Erd\H os-S\'os conjecture do not extend to give tights bounds in the hypergraph setting. A strengthening of the Erd\H os-S\'os conjecture due to the authors about tight lower bounds on the number of copies of a tree in a graph with average degree $d \ge t-1\ge 0$ remains open.

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Dhruv Mubayi, Jacques Verstraete. 2026-09-07. Kalai's Conjecture for Tight Trees. https://arxiv.org/abs/2609.08012

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