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

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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:

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

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