Monochrome Lines in the Plane
Jonathan Michael Borwein · Mathematics Magazine · 1979
Recently Tingley [3] has shown that, given any two disjoint, connected, compact sets, A and B, in the plane which do not both lie entirely on one line, one can find a line which passes through at least two points of one of these sets and misses the other. Such a line is said to be monochrome and the set it passes through is said to have a monochrome line. Tingley's theorem can be viewed as a variant of Motzkin's result (see [3] for discussion) that any two disjoint finite sets jointly spaning R2 (that is, which do not both lie on the same line) possess a monochrome line. In this note we first show that Tingley's conditions can be considerably weakened. Then we give conditions for uncountably many monochrome lines to exist. The following notational conventions will be useful: ab denotes the line through points a and b; [a, b] denotes the closed segment between a and b; P(b,B) denotes the pencil consisting of all lines through points b E B and a fixed b 4 B; co B denotes the convex hull of B, i.e., the smallest convex set containing B; ff denotes the closure of B; A B denotes the points of A which are not in B; and if L is a line, L and L denote the two closed half planes it generates.