Extremal Bounds for Three-Neighbour Bootstrap Percolation in Dimensions Two and Three

For $r\geq1$, the $r$-neighbour bootstrap process in a graph $G$ starts with a set of infected vertices and, in each time step, every vertex with at least $r$ infected neighbours becomes infected. The initial infection percolates if every vertex of $G$ is eventually infected. We exactly determine the minimum cardinality of a set that percolates for the $3$-neighbour bootstrap process when $G$ is a $3$-dimensional grid with minimum side-length at least $11$. We also characterize the integers $a$ and $b$ for which there is a set of cardinality $\frac{ab+a+b}{3}$ that percolates for the $3$-neighbour bootstrap process in the $a\times b$ grid; this solves a problem raised by Benevides, Bermond, Lesfari and Nisse [HAL Research Report 03161419v4, 2021].