Let there be a plane $P$ where all points $(x,y)$ are colored either $red$ or $blue$. Is it always possible to construct every regular $n$-gon where all the vertices are the same color?

Some observations and things (That might possibly be helpful?):

- There is always a line such that its endpoints and midpoint are the same colors. (I feel like this one could be useful for showing continuity when done an infinite number of times? idk)
- This is true for the triangle and square case. (Presumably also the rectangle case but I don't even know how to go about solving that. Also I feel like it should be easily provable with hexagons but I don't have the time right now - will update later)
- It is obvious that if there are only a finite amount of points of some color, then it is possible (I think).

Note: if you can't, a counterexample would be appreciated, as well as any other exceptions and reasons as to why.

Double Note: if you can, I would also like to know if you can just make a line of any length and any shape always.