On sets intersecting every square of the plane

We analyse properties of sets that contain at least one vertex of each square of the plane, in particular we study minimal elements (with respect to the subset relation) of the family $\mathcal A$ of sets with this property. We prove that minimal elements of $\mathcal A$ can avoid intersecting a bounded, dense, connected set of full outer measure (and thus of Hausdorff dimension 2). The motivation for this comes from Toeplitz’ conjecture – Every Jordan curve contains four points that are the vertices of a square – and the fact that the following is equivalent to Toeplitz’ conjecture: For all minimal sets $A$ in $\mathcal A$, the set $\mathbb R^2\setminus A$ does not contain a Jordan curve.