In the setting of the [Hadwiger-Nelson problem](https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_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 $|S|=\frak{c}$?



Subgraphs of $\mathbb{R}^2$ in the Hadwiger-Nelson problem
No. There is a finite $S$ with the same chromatic number. See [this](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory)).


