Connected plane sets which contain no nondegenerate connected simple graph

Abstract:

Open questions are answered by showing that, if for each vertical line $L$ in the plane, $1 < {k_L} \leq c$ is a cardinal number, there is a connected subset $H$ of the plane such that, if $L$ is a vertical line, then $L \cap H$ has cardinality ${k_L}$ and, if $g$ is a nondegenerate connected subset of $H$, then $g$ contains two points of some vertical line.