Yesterday I came up with a problem: Can we color each point of the plane with finitely many colors such that there doesn't exist any monochromatic regular polygonal?

But I found the problem is too hard for me(maybe someone could solve it for me, or the problem has been studied), so I considered the weaker form: Can we color each point of the plane with finitely many colors such that there doesn't exist any monochromatic equilateral triangle?

After some case discussion, one can prove that only two colors isn't enough, then I have no idea how to work on three colors.

If we use complex number, the problem becomes: Let $\mathbb{Z}[\omega]=\{a+b\omega\mid a,b\in\mathbb{Z},\;\omega=e^{\frac{2i\pi}{3}}\}.$ Can we color each elements of $\mathbb{Z}[\omega]$ such that there doesn't exist any monochromatic solution $(x,y,z)$ to the equation $x^2+y^2+z^2=xy+yz+zx$?

More generally, give a $n$-variable equation $f(x_1,\dots,x_n)$ and a infinite set $S$, can we color each elements of $S$ such that there doesn't exist any monochromatic solution $(x_1,\dots,x_n)$ to the equation $f(x_1,\dots,x_n)=0$?