Two sets of points of the plane $A$ and $B$ are said to be similar iff there is a bijection $f: A\rightarrow B$ and a constant $c_f\in\Bbb{R}^*_+$ such that $\forall X, Y\in A: \overline{XY} = c_f\cdot \overline{f(X)f(Y)}$, so, for example, every two regular $n$-agons are similar.
Let $n\in\Bbb{N}$ and $S$ be a finite set of points of the plane. Every point of the plane is colored with one of $n$ colors. Is there a monochromatic set $S'$ of points of the plane similar with $S$?
I know already that the statement is true for any $n$ if $|S|\le 3$ (if $|S| = 1$ or $2$ the statement is trivial; if $|S| = 3$, a small adaptation of the solution to this problem along with induction in $n$ will do the job).
I believe that the statement is true for any $n$ if $|S| = 4$, but I was not able to prove it.
What I'm interested in, in fact, is finding a counter example: For what finite coloring of the plane and what finite set of points of the plane is the statement false? How can we construct this set and this coloring?