The threshold for stacked triangulations

A stacked triangulation of a d-simplex o = {1, …, d + 1} (d ≥ 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, that is, for which p does the random simplicial complex Y ∼ Y_d(n, p) contain the faces of a stacked triangulation of the d-simplex o, with its internal vertices labeled in [n]. In the language of bootstrap percolation in hypergraphs, it pertains to the threshold for K^d+1d+2, the (d + 1)-uniform clique on d + 2 vertices. Our main result identifies this threshold for every d ≥ 2, showing it is asymptotically (α_dn)^−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.