If $G =(V,E)$ is a simple, undirected graph (finite or infinite), and $\kappa \neq \emptyset$ is a cardinal, we say that the complete graph $K_\kappa$ is a minor of $G$ if there is a collection ${\frak S}$ of connected and pairwise disjoint subsets of $G$ such that

1. $|{\frak S}| = \kappa$, and
2. whenever $S\neq T\in {\frak S}$, then there is $e\in E$ such that $(S\cap e) \neq \emptyset \neq (T\cap e)$.

The Hadwiger-Nelson graph has $\mathbb{R}^2$ as its ground set, and two elements $x,y\in\mathbb{R}^2$ form an edge if and only if their Euclidean distance equals $1$.

Question. Is the set of cardinals $\kappa \neq \emptyset$ such that $K_\kappa$ is a minor of the Hadwiger-Nelson graph finite? If yes, what is its greatest member?

Note. If the Hadwiger conjecture is correct, then the set asked for in the question is a super-set of $\{1,2,3,4,5\}$.

If $G =(V,E)$ is a simple, undirected graph (finite or infinite), and $\kappa \neq \emptyset$ is a cardinal, we say that the complete graph $K_\kappa$ is a minor of $G$ if there is a collection ${\frak S}$ of connected and pairwise disjoint subsets of $G$ such that

1. $|{\frak S}| = \kappa$, and
2. whenever $S\neq T\in {\frak S}$, then there is $e\in E$ such that $(S\cap e) \neq \emptyset \neq (T\cap e)$.

The Hadwiger-Nelson graph has $\mathbb{R}^2$ as its ground set, and two elements $x,y\in\mathbb{R}^2$ form an edge if and only if their Euclidean distance equals $1$.

Question. Is the set of cardinals $\kappa \neq \emptyset$ such that $K_\kappa$ is a minor of the Hadwiger-Nelson graph finite? If yes, what is its greatest member?

Note. Since the Hadwiger conjecture is correct for $k\leq 6$, the set asked for in the question is a super-set of $\{1,2,3,4,5\}$, because it is known that the chromatic number of the Hadwiger-Nelson graph is a member of $\{5,6,7\}$.