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In article <1993Jan9.163703.24700@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.)

Stop there.  The assertion is false.  For a counter example, color all
points with rational coordinates red and those with irrational coordinates
blue.

For a simple proof, consider this figure (equilateral triangular net);:

                 0
                1 2
               3 4 5
              6 7 8 9

This figure contains a same-colored equilateral triangle.  Proof:
Suppose it does not.   Then 1-5-7 contains both colors; one is the same
as the color of 4.  Without loss of generality we can suppose that 1 and
4 are red and that at least one of 5 and 7 are blue, which gives us:


                 0
                R 2
               3 R 5
              6 7 8 9

Then it follows that 2 and 3 must be blue and, to keep 2-3-8 multi-colored,
8 must be red.  In turn this implies that 5 and 7 are both blue, giving us

                 0
                R B
               B R B
              6 B R 9

Now from triangle 3-6-7 6 must be R, from triangle 2-7-9 9 must be R,
and from 0-3-5 0 must be R, so that 0-6-9 is all R, contrary to the
assumption.  Q.E.D.

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