Let $x,y\in \mathbb{Z}^\omega$ and let $x,y\in\mathbb{Z}^\omega$ form an edge if there is $i\in\omega$ such that $|x\_i - y\_i|=1$ and $ x\_k = y\_k$ for all $k\in \omega\setminus\{i\}$.


$K\_\omega$, the complete graph on $\omega$ points, is a [minor](https://dominiczypen.wordpress.com/2018/06/02/a-definition-of-minor-in-graph-theory/) of $\mathbb{Z}^\omega$ (see [this post](https://mathoverflow.net/a/301842/8628)).


**Question.** Is it possible that there is a cardinal $\lambda>\omega$ such that $K\_\lambda$ is a minor of $\mathbb{Z}^\omega$?



Large complete minors of $\mathbb{Z}^\omega$
If I understand the definition correctly, the answer is no. 


If $K\_\lambda$ is a minor of $G$, then there is a 1-1 map from $K\_\lambda$
into the family of (nonempty) connected subsets of $G$, such that any two images are in the same connected component of $G$, but disjoint. So there must be a component of size $\ge \lambda$. 


If you take $G$ to be $\mathbb Z^\omega$, then every point in $G$ has only countably many neighbors, so all connected components are countable. 


