Is there an infinite, simple, undirected graph $G=(V,E)$ such that there is $n\in\mathbb{N}$ with the following properties? 


1. $K\_n$ is a [minor](https://en.wikipedia.org/wiki/Graph_minor) of $G$, but $K\_{n+1}$ is not a minor of $G$, and
2. if $F$ is a finite subgraph of $G$, then $K\_n$ is not a minor of $F$.


(Note: I use "compactness" in the title because this question has certain parallels with the compactness result of [Erdös-De Bruijn](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory)).)



Compactness of Hadwiger number
Assume $K\_n$ is a minor of $G$. Each vertex of $K\_n$ corresponds to a connected subset of vertices of $G$ as it can only have been obtained by contracting edges. Each edge of $K\_n$ can only have been derived from a path through $G$, by contracting edges. 
Now create $F$ as follows: collect the edges of the $n(n-1)/2$ paths and the vertices incident to these, this gives of finite subsets of the aforementioned connected sets that gives us the vertices of $K\_n$. Add finitely many edges to make these finite sets connected. The resulting graph has $K\_n$ as a minor.
This means that condition 2 in the question can not be met. 


