Coloring of $\mathbb{R}^3$ into 3 colors 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?

You have proven that for any distance $\delta$, we can find two points at that distance whose colors match.  What you haven't proven (and what is requested) is that we can always find two points at that distance that are both a particular color (red, for example).  Your construction leaves open the possibility that for (say) $\delta = 1$ all pairs thus constructed are blue, while for $\delta = 2$ all pairs thus constructed are red.
EDIT:  I'm not entirely sure what the logic is in your revised proof;  the concluding sentences are rather unclear to me.  However, it appears to assume that the vertices of any tetrahedron include all three colors, but this is not necessarily the case.
