Connected plane sets which contain no nondegenerate connected simple graph

Abstract

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