Translation Orders on Grids

Abstract

Let $S$ be a grid in $\mathbb{Z}^d$. We wish to order $S$ so that the way two points compare depends only on the offset between them. Does each such order arise from a linear function? Bhattacharya conjectured, while Ren and Huang confirmed, that in two dimensions the answer is yes. On the other hand, wild orders do exist when $d \ge 4$. We settle the remaining case, by showing that the answer is yes in three dimensions as well.