We coloring the plane R^2 in two different colors.
1. Prove that there is 2 points that in 1 unit distance from each other.
2. Prove that exist a rectangle that his edge points colored with the same color.
Point 1. Hint: can you find three equidistant points on the plane?
Point 2. Hint: Suppose we found 6 points located like this:
Let be the color of and assume that the triple equals the triple (pointwise). Can you find the required rectangle?
Now, can you find 6 points with this property?