TITLE: Coloring of $\mathbb{R}^3$ into 3 colors
QUESTION [6 upvotes]: Every point of three-dimensional space is colored red, green, or blue. Prove that one of the colors attains all distances, meaning that any positive real number represents the distance between two points of this color.
My proof: Suppose by contradiction that $\exists \delta>0$ such that for any $x,y\in \mathbb{R}^3$ with $d(x,y)=\delta$ points $x$ and $y$ have different colors. Let's consider the tetrahedron in $\mathbb{R}^3$ with base $A_1A_2A_3$ and upper vertex $A_4$. Suppose $A_1$ is colored into red,$A_2$ is colored into green then $A_3$ is colored into blue. Then $A_4$ should be colored into one of the colors blue, red, or green but due to $d(A_4,A_1)=d(A_4,A_2)=d(A_4,A_3)=\delta$ the coloring of point $A_4$ to red,blue, or green is impossible.
EDIT: Let's prove that RED attains all distances.Suppose by contradiction that $\exists \delta>0$ such that for any $x,y\in \mathbb{R}^3$ with $d(x,y)=\delta$ points $x$ and $y$ not both RED. Let's consider the regular tetrahedron in $\mathbb{R}^3$ with base $A_1A_2A_3$ and upper vertex $A_4$. WLOG suppose $A_1$ is colored into red,$\ A_2$ is colored into green then $A_3$ is colored into blue. Then $A_4$ should be colored into one of the colors blue, red, or green but due to $d(A_4,A_1)=d(A_4,A_2)=d(A_4,A_3)=\delta$ but the coloring of point $A_4$ to red,blue, or green is impossible. Here we get contradiction.
Right?

REPLY [2 votes]: Suppose there is no color which attains all distances. This means there exists a distance $r$ not attained by any two red points, a distance $b$ not attained by any two blue points, and same for $g$ with green points. WLOG $r\ge b\ge g$. 
Let $x$ be any red point*. Let $S_x$ be the (surface of the) sphere of radius $r$ centered at $x$. Every point in $S_x$ must be blue or green.
Let $y$ be any blue point in $S_x$**. Let $S_y$ be the sphere of radius $b$ centered at $y$. Since $b\le r$, the intersection of $S_x$ and $S_y$ is a circle. Every point in that circle must be green. 
After drawing a picture, you should be able to convince yourself that the diameter of this circle is at least $b$; since $b\ge g$, there will exist two points on the circle whose distance is $g$, which contradicts the supposed fact that no two green points are at distance $g$. 
*Here we are assuming that there exists a red point. What if there are no red points? This is left as an exercise to the reader. 
**What do you do if no blue point exists?