Unable to solve:Coloring vertices in higher dimensions

I had been working on this idea, and I am unable to solve it

The problem I pick up is as follows:

In n dimensions you does the following things:
You take up the n dimensional square
It will have 2^n vertices
You find all the possible triangle that are present in the square
Now you color them with Green and Red colors following these rules:
No triangle have all of it's vertices of the same color
50% of the vertices are colored red and rest of them are green

Now what is the minimum possible dimension (minimum possible n) in which you can't color the all the vertices without breaking the coloring rules?

Note:The idea of n dimensional square is similar to the n dimensional square idea for the Graham's number

#Logic

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$