# Three-colored plane

Let $\Pi$ is cartesian plane with the usual topology. $A, B, C$ are pairwise disjoint subsets $\Pi$, $A\cup B\cup C = \Pi$. Each of these sets is dense in $\Pi$: $\overline{A}=\Pi$, and $\overline{B}=\Pi$, and $\overline{C}=\Pi$. Is it always possible to find a straight line that contains points $a,b,c$ of these sets (for some $a\in A, b\in B, c\in C)$?

It is always possible if one of these sets contains a line segment.

• Thank you for your attention. I found the relevant topic here: mathoverflow.net/questions/134475/… – grizzly Jan 15 '15 at 9:42
• A neat solution (linked the mathoverflow question) is here It would be easy to bring in, but is marked copyright. – Ross Millikan Jan 15 '15 at 17:15
• @RossMillikan: Nice solution. – copper.hat Jan 15 '15 at 20:18