## Abstract

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}^{+1}^{d}+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_{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.

Original language | American English |
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Pages (from-to) | 16296-16335 |

Number of pages | 40 |

Journal | International Mathematics Research Notices |

Volume | 2023 |

Issue number | 19 |

DOIs | |

State | Published - 1 Oct 2023 |

### Bibliographical note

Publisher Copyright:© The Author(s) 2022.