Points in the plane are each colored with one of two colors: red or blue. Prove that, for a given distance d, there always exist two points of the same color at the distance d from each other.

Solution to Problem 1

Select a point O. With O as the center, draw a circle of radius d. There are just two possibilities:

1. The circle thus drawn contains a point of the same color as O. If that's the case we are finished.
2. All points of the circle have a color different from the O's. Then any chord of length d connects two points of the same color.

A question may be asked by a student or suggested by a teacher, Is it always the case that in a circle of radius d there exists a chord of length d? The answer may at first seem obvious but, depending on a student's level, may eventually prove insurmountable. A discussion here may be extremely useful.

For the lower grades, it's possible to simply stipulate existence of such a chord. For more advanced students, a better axiom would be the one that asserts that two circles, each passing through the center of another, intersect at two points.

Points in the plane are each colored with one of three colors: red, green, or blue. Prove that, for a given distance d, there always exist two points of the same color at the distance d from each other.

Solution to Problem 2

As before, choose an arbitrary point O. Let it be red. With the center at O and radius equal twice the altitude of an equilateral triangle with side d draw a circle. As in Problem 1, if all the points on that circle are red, there is nothing to prove (Adapt the axiom from Problem 1 to the case of circles having distinct radii). Let there be a point P of a different color, say, green. Draw two circles with radius d: one centered at O, another at P. They intersect at two points A and B such that both triangles OAB and PAB are equilateral. If A is either red or green, we are finished. Assume it's blue. Then B lies at the distance d from three points O, P, and A, all of different colors. One of these has the same color as B.

While solving the two problems above we had to stipulate supporting axioms. Formulate a most general axiom and check that the original ones can be derived from it.

Points in the plane are each colored with one of two colors: red or blue. The set of distances between the blue points is blue and the set of distances between the red points is red. Prove that either one or the other contains all the positive reals.

Solution to Problem 3

Assume to the contrary that the blue set lacks distance b and the red set lacks distance r. Choose a blue point B and form an isosceles triangle ABC with AB = BC = b and AC = r. Then either both A and C are red or one of them is blue. If they are both red we have two red points at distance r. If one of them blue, we have to blue points at distance b.