Steinhaus Sets and Jackson Sets
  We prove that there does not exist a subset of the plane S that meets every isometric copy of the vertices of the unit square in exactly one point. We give a complete characterization of all three point subsets F of the reals such that there does not exists a set of reals S which meets every isometric copy of F in exactly one point. A finite set X in the plane is Jackson iff for every subset S of the plane there exists an isometric copy Y of X such that Y does not meets S in exactly one point. These results are related to the open problem:   Q. (Steve Jackson) Is every finite set X in the plane of two or more points Jackson? 
Su Gao, Arnold W. Miller, and William A. R. Weiss