Let $n>1$ be an integer. We say that two points $(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in\mathbb{Z}^n$ are a member of the edge set $E_n$ if and only if $$\sum_{i=1}^n|x_i-y_i| = 1.$$

**Question.** Given an integer $n>1$, is there a maximal integer $m(n)$ such that the complete graph $K_{m(n)}$ is a minor of $(\mathbb{Z}^n, E_n)$? If yes, it would also be nice to know an explicit formula for $m(n)$.

**Remark.** I believe that for $n=2$ there is such a maximal integer, and $m(2) \in \{3,4\}$. I am fairly certain that $m(2) >4$ would allow to construct a counterexample to the 4-color theorem.)

**Edit.** In the comment section, Wojowu shows that $m(2)=4$.