🚀 go-pugleaf

>In article <1993Jan9.030824.13624@Csli.Stanford.EDU> hiraga@Csli.Stanford.EDU (Yuzuru Hiraga) writes:
>>In article <1993Jan8.193314.3289@zip.eecs.umich.edu> kanad@quip.eecs.umich.edu (Kanad Chakraborty) writes:
>>
>>A plane is totally colored with two colors.
>>* Show that there exist three points of same color that
>>  (when connected) form an equilateral triangle.
>
>Proof. WLOG, let the colors be red and blue.
>Choose a point A in the colored plane. Suppose, WLOG, that 
>A is colored blue. Find the (or a) point colored red _nearest_ to A 
>(Euclidean distance-wise). (This is always possible because both
                             ^^^^^^^^^^^^^^^^^^^^^^^
>colors have been used.) Call this point B. Suppose the
                    [stuff deleted]
>Kanad
 

It is NOT always possible to find such a nearest red point.
Suppose the plane is colored as follows:

   (x,y) is colored blue if x is rational
                 and red if x is irrational.

or a simpler coloring which shows this is the following:

   The point (0,0) is colored blue and every other point
   is colored red.

   This blue point clearly has no closest red point.


Bob Murphy (rmurphy@aludra.usc.edu)