Three-Neighbour Bootstrap Percolation in Thin Three-Dimensional Grids
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911158244474880 |
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| author | Dolphin, Will Dukes, Peter J. |
| author_facet | Dolphin, Will Dukes, Peter J. |
| contents | We improve the status of the problem of determining minimum-sized percolating sets in $a \times b \times c$ grids under the $3$-neighbour process. Using several new constructions, we show that optimal percolating sets exist whenever $\min(a,b,c) \ge 7$. As an important step toward this, we also show that all grids with $\min(a,b,c) \ge 4$ have a percolating set whose size exactly achieves the lower bound $(ab+ac+bc)/3$ whenever this value is an integer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13589 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Three-Neighbour Bootstrap Percolation in Thin Three-Dimensional Grids Dolphin, Will Dukes, Peter J. Combinatorics 05C57, 05C35, 82B43 We improve the status of the problem of determining minimum-sized percolating sets in $a \times b \times c$ grids under the $3$-neighbour process. Using several new constructions, we show that optimal percolating sets exist whenever $\min(a,b,c) \ge 7$. As an important step toward this, we also show that all grids with $\min(a,b,c) \ge 4$ have a percolating set whose size exactly achieves the lower bound $(ab+ac+bc)/3$ whenever this value is an integer. |
| title | Three-Neighbour Bootstrap Percolation in Thin Three-Dimensional Grids |
| topic | Combinatorics 05C57, 05C35, 82B43 |
| url | https://arxiv.org/abs/2509.13589 |