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](https://dominiczypen.wordpress.com/2018/06/02/a-definition-of-minor-in-graph-theory/) 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](https://en.wikipedia.org/wiki/Four_color_theorem).)


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



Complete minors of the grid graphs $\mathbb{Z}^n$
Either I am missing something or for $n>2$ you have $m(n)=\infty$. It is enough to show this for $n=3$. Choose any $m$ and a set of edges in $E\_3$ which is the union of the following three sets:


* $\{(i,j,0),(i,j+1,0)\}\mbox{ such that } 1\leq i\leq m\mbox{ and } 1\leq j < m$
* $\{(i,j,1),(i+1,j,1)\}\mbox{ such that } 1\leq i < m\mbox{ and } 1\leq j\leq m$
* $\{(i,i,0),(i,i,1)\}\mbox{ such that } 1\leq i\leq m$


Then the subgraph determined by these edges has $m$ connected components and any two of them are "connected" to each other in your sense.


