Set Theory of the Plane

I plan to present the Jackson-Mauldin solution to the Steinhaus problem. In the 1950s Steinhaus asked if there exists a subset S of the plane which meets every isometric copy of the integer lattice, Z×Z, in exactly one point. I also plan to cover other results which involve the set theoretical properties of the plane. Davies has shown that the plane can be partitioned into countably many pieces none of which contains two pairs of points the same distance apart iff the continuum hypothesis is true. On the other hand Schmerl has shown that without any extra set theoretical hypotheses the plane can always be partitioned into countably many pieces so that no piece contains the vertices of an isoceles triangle.