Help Jitendra Attack a 3-D Coloring Problem

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AlbertEinstein
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Hi everybody, I am unable to tackle this problem, and don't know how to attack it. can someone please help me how to attack the following problem.

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.

please give some hints.
thank you.
Jitendra
 
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I don't know how to approach it...but know that it is based on the concept of denumerability of real numbers. I don't want complete help but just few hints so that i can do it myself.please help.

thanks
 
AlbertEinstein said:
I don't know how to approach it...but know that it is based on the concept of denumerability of real numbers. I don't want complete help but just few hints so that i can do it myself.please help.

thanks

The real numbers aren't denumerable though. That is they are not countable, or there exists no bijection between the reals and the natural numbers.
 
I don't think there are any advanced math tricks that'll help you. You just need to think it through for a while. To get you started, the first one is true if the space is 2D. Can you prove this?