What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all the euclidean distances from X to points of S are rational?

Can we do better then $3\leq K \leq 2^{\aleph_0}$?

What can be said about K if all the points of S must be at integer distance from eachother, or algebraic distance?