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. (edit: I have since realized this is not true. Say we wanted to be able to make a straight line of length 1. Consider a $red$ plane. Now we can color a square grid $blue$ s.t the side length of each square is 0.5 units. Then, at each intersection of the squares, replace the vertex with a hollow $blue$ circle of appropriate radius, where the points inside the circle are $red$. We cannot make a straight line segment of any color over a certain length.)