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The threshold for stacked triangulations

A \emph{stacked triangulation} of a $d$-simplex $\mathbf{o}=\{1,\ldots,d+1\}$ ($d\geq 2$) is a triangulation obtained by repeatedly subdividing a $d$-simplex into $d+1$ new ones via a new vertex (the case $d=2$ is known as an Appolonian network). We study the occurrence of such a triangulation in the Linial--Meshulam model, i.e., for which $p$ does the random simplicial complex $Y\sim \mathcal{Y}_d(n,p)$ contain the faces of a stacked triangulation of the $d$-simplex $\mathbf{o}$, with its internal vertices labeled in $[n]$. In the language of bootstrap percolation in hypergraphs, it pertains to the threshold for $K_{d+2}^{d+1}$, the $(d+1)$-uniform clique on $d+2$ vertices. Our main result identifies this threshold for every $d\geq 2$, showing it is asymptotically $(α_d n)^{-1/d}$, where $α_d$ is the growth rate of the Fuss--Catalan numbers of order $d$. The proof hinges on a second moment argument in the supercritical regime, and on Kalai's algebraic shifting in the subcritical regime.