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Q. Suppose colour every point in 3-D space is assigned one of the three colours- red,green,blue.Can i conclude the following???:

1)there must exist a right triangle which has three of the vertices of same colour.

2)there must exist an equilateral triangle which has all its vertices of same colour.

3)the problems 1 and 2 with the additional fact that there exist infinitely many such in any region of space.

4)there must exist a monochromatic line.

5)there must exist a monochromatic circle.

plz give some hints.

thank you.

Jitendra