Upper bounds on the running time of bootstrap percolation
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866915955377963008 |
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| author | Liu, Weichan Nie, Xiangxiang Piga, Simón Schülke, Bjarne |
| author_facet | Liu, Weichan Nie, Xiangxiang Piga, Simón Schülke, Bjarne |
| contents | For $k$-graphs $F$ and $H_0$ the $F$-bootstrap percolation process (or $F$-process) starting with $H_0$ is a sequence $(H_i)_{i\geq0}$ of $k$-graphs such that $H_{i+1}$ is obtained from $H_i$ by adding all those $e\in V(H_0)^{(k)}\setminus E(H_i)$ as edges that complete a new copy of $F$. The running time of this $F$-process, denoted by $M_F(H_0)$, is the smallest $i$ with $H_i=H_{i+1}$. Bollobás proposed the problem of determining the maximum running time for $n\in\mathbb{N}$, i.e., $M_F(n)=\max_{\vert V(H_0)\vert=n}M_F(H_0)$. Although this problem has received a lot of attention recently, until now the best known upper bound for $M_{K_t}(n)$, with $t\geq5$, was the trivial bound $\binom{n}{2}$.
Here we provide the first non-trivial upper bound for this problem by showing that $$M_{K_t}(n)\leq\Big(\frac{t-3}{t-2}+o(1)\Big)\binom{n}{2}$$ holds for every integer $t\geq 3$. In fact, we prove the following more general result. For every $k\geq2$, every $k$-graph $F$, and every $e\in E(F)$ we have $M_F(n)\leq\big(π(F-e)+o(1)\big)\binom{n}{k}$, where $π$ is the Turán density. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22630 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Upper bounds on the running time of bootstrap percolation Liu, Weichan Nie, Xiangxiang Piga, Simón Schülke, Bjarne Combinatorics For $k$-graphs $F$ and $H_0$ the $F$-bootstrap percolation process (or $F$-process) starting with $H_0$ is a sequence $(H_i)_{i\geq0}$ of $k$-graphs such that $H_{i+1}$ is obtained from $H_i$ by adding all those $e\in V(H_0)^{(k)}\setminus E(H_i)$ as edges that complete a new copy of $F$. The running time of this $F$-process, denoted by $M_F(H_0)$, is the smallest $i$ with $H_i=H_{i+1}$. Bollobás proposed the problem of determining the maximum running time for $n\in\mathbb{N}$, i.e., $M_F(n)=\max_{\vert V(H_0)\vert=n}M_F(H_0)$. Although this problem has received a lot of attention recently, until now the best known upper bound for $M_{K_t}(n)$, with $t\geq5$, was the trivial bound $\binom{n}{2}$. Here we provide the first non-trivial upper bound for this problem by showing that $$M_{K_t}(n)\leq\Big(\frac{t-3}{t-2}+o(1)\Big)\binom{n}{2}$$ holds for every integer $t\geq 3$. In fact, we prove the following more general result. For every $k\geq2$, every $k$-graph $F$, and every $e\in E(F)$ we have $M_F(n)\leq\big(π(F-e)+o(1)\big)\binom{n}{k}$, where $π$ is the Turán density. |
| title | Upper bounds on the running time of bootstrap percolation |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.22630 |