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 $|S|=\frak{c}$?