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:
$\displaystyle p_{11}$ $\displaystyle p_{12}$
$\displaystyle p_{21}$ $\displaystyle p_{22}$
$\displaystyle p_{31}$ $\displaystyle p_{33}$
Let $\displaystyle c_{ij}$ be the color of $\displaystyle p_{ij}$ and assume that the triple $\displaystyle (c_{11}, c_{21}, c_{31})$ equals the triple $\displaystyle (c_{12}, c_{22}, c_{32})$ (pointwise). Can you find the required rectangle?
Now, can you find 6 points with this property?