A circular spintop is colored in blue, red and green. Whenever the spintop is rotated 120 degrees, the pattern of colors looks exactly the same, only that blue becomes red, red becomes green and green becomes blue, like this:
Prove that there is a blue segment which meets a red segment in one side and a green segment in the other side.
I managed to prove this when the number of segments of each color is finite.
First, because of the rotating condition, it is obvious that all 3 colors must be present in the coloring (for example, if there is a blue point, then there must also be a red point and a green point). Moreover, the number of segments of each color is equal.
Suppose there are $k$ segments of each color. Then the number of endpoints of segments of each color is $2k$. Suppose by contradiction that every blue segment meets only red segments. This means that every endpoint of a blue segment must meet an endpoint of a red segment. But then there is no room for any green segment - a contradiction.
My questions are:
- Is this proof correct?
- Is the statement true also when the number of segments of each color is infinite?