Question 72
Consider $n$ collinear points $P_{1}, \ldots, P_{n}$ and all circles with diameter $P_{i} P_{j}$ for $1 \leq i\lt j \leq n$. Each of these circles is coloured with one of $k$ colours. Such a set of coloured circles is called an $(n, k)$-tangle. A monochromatic figure eight consists of two circles of the same colour that touch each other externally. Show that every $(n, k)$-tangle contains a monochromatic figure eight if and only if $n\gt 2^{k}$.