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nt.quantity idea – Dense units in $Bbb{R}^2$ with rational distance

We convene a subset $Ssubset Bbb{R}^2$ *rationally distanced* if all $s_1,s_2 in S$ have rational Euclidean distance.

The Erdos-Ulam surmise asks if there’s a dense subset of $Bbb{R}^2$ which is rationally distanced. Inspired by this, I got here up with the next toy drawback.

A subset $X subset Bbb{R}^2$ is *good* if there exists a rationally distanced $S subset Bbb{R}^2setminus overline{X}$ such that every $xin X$ is a restrict level of $S$.

I’ve three questions:

- Is $[0,1]$ good?
- If Question 1 is disloyal, does this suggest the Erdos-Ulam surmise to breathe disloyal?
- Is there a very good clique which isn’t bounded?

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