Does there exist a set of $n$ points $p_1,p_2,...,p_n$ in the plane, all at mutual integer distances to each other, and an $e>0$, such that the following statement holds:
For all $a,b$ with $a^2+b^2<e$, there exists an integer $i$, with $1\leq i\leq n$, such that the distance from $(a,b)$ to $p_i$ is irrational.
What is the least such $n$?