Question 60
Let $k \geqslant 1$ be an integer. What is the smallest integer $n$ such that, however $n$ points are placed in the plane, it is possible to choose a subset $S$ consisting of $k$ of these points satisfying "for every pair $P, Q$ of points of $S$, the distance between $P$ and $Q$ is less than or equal to 2" or "for every pair $P, Q$ of points of $S$, the distance between $P$ and $Q$ is strictly greater than 1."