Conditional three-way coloring of a cicular disc with unit radius

alghosh · March 2nd, 2015, 9:32 am

This is some Olympiad problem ( I don't know the source ) :You have three colors - say Green, Red and Yellow. Is it possible to color (using all three colors) the unit radius circular disk such that distance between any two points of the same color is never one unit!We can have a tricky solution( I have one ) however can we solve the general case , i.e for n-dimensional sphere and (n+1) disjoint set such that this holds or not?

cm27874 · March 10th, 2015, 10:19 am

SPOILER: Hadwiger-Nelson problem. Both the Moser Spindle and Solomon W. Golomb's ten-vertex graph (cf. Wikipedia) fit into the unit radius circular disk. Both require four colors.

alghosh · March 11th, 2015, 6:50 am

Thanks cm27874 for the info. Yes I got the same Solomon W. Golomb's 10- vertex graph. However Moser Spindle is quite interesting.

wileysw · March 13th, 2015, 4:58 am

algosh, i am recommending this book by Soifer. there are so many interesting anecdotes about the history of this so-called "chromatic number of the plane", though there are painstakingly long chapters about van der waerden that one might not find as fascinating...

And2 · June 2nd, 2015, 6:08 pm

The original (Olympiad) problem... did you mean coloring of the edge of the 2d disc (i.e, circle), or indeed the whole disc? (I've given up trying to solve the disc problem and Google search gave something similar but for the circle) ... and as stated the problem definitely does not have solution, as the center of the disk and its edge must be of different colors already ..