# Colourful space

Hi everybody, I am unable to tackle this problem, and don't know how to attack it. can someone plz 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.

plz give some hints.
thank you.
Jitendra

cristo
Staff Emeritus
Show some work! What do you think?

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.plz help.

thanks

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.plz 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.

StatusX
Homework Helper
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?