Kalai's Conjecture for Tight Trees

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 $|E(H)|\le (t-1)|\mathrm{sh}H|/r$, where $\mathrm{sh}H$ 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-Sós Conjecture. The proof was found by GPT-6 Astra, extending its method of proof for the Erd\H os-Sós conjecture to the hypergraph setting. It is noteworthy that previous proofs of special cases of the Erd\H os-Sós conjecture do not extend to give tights bounds in the hypergraph setting. A strengthening of the Erd\H os-Sós 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.