Let $a_{n},\ n\ge 1$ be a sequence of positive reals such that $\left(\sum_{n\ge 1}a_{n}\right)^{2}<2\pi\ (*)$. For each $n\ge 1$, let $\mathcal C_{n}$ be the circle centered at $O$ with radius $\sum_{i=1}^{n}a_{i}$. Because of $(*)$, we can find points $x_{i,j}\in\mathcal C_{i},\ i,j\ge 1$ such that for all $i,j\ge 1$ we have $\mathcal C(x_{i,j})=\mathcal C_{i+j}$. We now forget about all the other points, and work only with the matrix $M=(x_{i,j})$. Suppose we use $n$ colors. There must be one, $c_{1}$, which appears infinitely many times on the first row of $M$, in, say, points $x_{1,j_{1}},x_{1,j_{2}},\ldots$. Then $c_{1}$ cannot appear on the lines $j_{k}+1$, $ k\ge 1$. Next, there is a color $c_{2}$ which appears infinitely often among the points $x_{j_{1}+1,j_{k}-j_{1}}$, $k\ge 2$. But then $c_{2}$ cannot appear on the lines $j_{k}+1$ for such $k$. Repeating this procedure, we reach a stage where we have a row of $M$ (infinitely many actually) on which none of our $n$ colors $c_{1},\ldots,c_{n}$ can appear. This is a contradiction.