Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 31J / b / i / Solution

Choose $2m$ ordered points. Every subset of them has at most $m$ maximal consecutive runs, and those runs can be covered by at most $m$ closed intervals. Thus the points are shattered. On $2m+1$ ordered points, however, the alternating labelinghas $m+1$ separated positive runs and cannot be realized by a union of $m$ intervals. Therefore