A Discrete Geometrical Gem
4. Generalizations and adding color
Motzkin showed (with a very nice argument) that if one has two finite sets X and Y in the plane which have no points in common, the points in X or Y all lie on a line or there exist two points of one of these sets that determine a line that does not intersect the other set. Sometimes the condition that points do not all lie on a single line is stated as the points in X or Y "span" the plane. If we think of the points of X as being red and those of Y being blue, we can interpret this theorem as saying that if we have two finite sets of points which are of different colors, then if the points do not all lie on a line, there is a monochromatic line of one of the two colors.
Welcome to the
These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.
Search Feature Column
Feature Column at a glance