Subgraphs of $\mathbb{R}^2$ in the Hadwiger-Nelson problem

In the setting of the Hadwiger-Nelson problem, two points of $\mathbb{R}^2$ form an edge if and only if their distance is $1$. The resulting graph $G$ has chromatic number $\chi(G)\in \{4,5,6,7\}$ and has uncountably many connected components.

If we take $S\subseteq \mathbb{R}^2$ and regard it as an induced subgraph of $\mathbb{R}^2$ with the edge set described above such that $\chi(S)=\chi(G)$, does it follow that $|S|=\frak{c}$?

• The graph $G$ is connected (e.g. any two points can be connected by a chain of length $2$ if they are at distance $<1$). Despite the accepted answer, your question still makes sense in axiomatic theories without choice. – Benoît Kloeckner Apr 26 '16 at 10:03

No. There is a finite $S$ with the same chromatic number. See this.

• This assumes the axiom of choice, which should be mentioned given the dependency of related problems to the chosen axioms. – Benoît Kloeckner Apr 26 '16 at 10:05