Let $S$ be a set of points in a plane $P$, having the following property: for any point $X \in P$ there is at least one point $M \in S$ so that the distance $|XM|$ is rational. Find the minimum cardinal of such a set.
First, let's notice there is such a set, by taking $S=P$. Also if $S$ is a line in $P$. Someone claimed there is finite set $S$ having the required property. Does anyone have an idea how to solve this?
UPDATE
A related question here.
$S$ must have cardinality $\mathfrak c$.
The assumed property is that $$ P = \bigcup_{(M,d)\in S\times\mathbb Q} \bigl\{ X\in P\bigm| |XM|=d \bigr\} $$ The sets in the union on the right are circles, and each circle intersects the $x$-axis in at most two points. Since the $x$-axis has $\mathfrak c$ points in it, it can only be totally within the union if there are at least $\mathfrak c$ circles. However if $S$ is infinite (and finitely many circles certainly won't cover the plane), then $|S\times\mathbb Q|=|S|\times\aleph_0 = |S|$, so $|S\times\mathbb Q|\ge\mathfrak c$ implies $|S|\ge\mathfrak c$.
On the other hand, there are only $\mathfrak c$ points that can be in $S$.
In particular, "the points of a line having rational coordinates" won't do -- there are too few of them. (Not to mention that some lines, such as $x+y=\sqrt2$, contain no points with rational coordinates).