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We say that a finite, simple, undirected graph $G=(V,E)$ is $\mathbb{R}^2$-realizable if there is an injective map $\varphi:V\to \mathbb{R}^2$ such that for $v\neq w \in V$ we have $\{v,w\} \in E$ if and only if $|\varphi(v)-\varphi(w)| < 1$ where $|\cdot|$ denotes the Euclidean distance.

What is an example of a finite graph that is not $\mathbb{R}^2$-realizable?

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    $\begingroup$ Those are precisely the unit disk graphs (to replace the inequality by a strict one, scale the graph by a factor slightly smaller than $1$). $\endgroup$ – Wojowu Feb 12 at 15:29
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How about the star $S_6$? To realize this graph in the way you describe, you would have to map the center point to some $c \in \mathbb R^2$; then you would need to map the $6$ other points of $S_6$ inside the unit circle around $c$, but all at distance $\geq\!1$ from each other. This is impossible.

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