Three-Neighbour Bootstrap Percolation in Thin Three-Dimensional Grids

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 $(ab+ac+bc)\mathrm{/}3$ whenever this value is an integer.