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arXiv · 2209.02015

Long running times for hypergraph bootstrap percolation

Alberto Espuny Díaz·Barnabás Janzer·Gal Kronenberg·Joanna Lada
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Abstract

Consider the hypergraph bootstrap percolation process in which, given a fixed $r$-uniform hypergraph $H$ and starting with a given hypergraph $G_0$, at each step we add to $G_0$ all edges that create a new copy of $H$. We are interested in maximising the number of steps that this process takes before it stabilises. For the case where $H=K_{r+1}^{(r)}$ with $r\geq3$, we provide a new construction for $G_0$ that shows that the number of steps of this process can be of order $\Theta(n^r)$. This answers a recent question of Noel and Ranganathan. To demonstrate that different running times can occur, we also prove that, if $H$ is $K_4^{(3)}$ minus an edge, then the maximum possible running time is $2n-\lfloor \log_2(n-2)\rfloor-6$. However, if $H$ is $K_5^{(3)}$ minus an edge, then the process can run for $\Theta(n^3)$ steps.

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Alberto Espuny Díaz, Barnabás Janzer, Gal Kronenberg, Joanna Lada. 2022-09-05. Long running times for hypergraph bootstrap percolation. https://arxiv.org/abs/2209.02015

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