Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 31L / b / ii / Solution

The choice $\delta =1$ labels an interval positively and its complement negatively, while $\delta =-1$ reverses those labels. On three ordered points, every subset is either a consecutive block or the complement of a consecutive block, so all eight labelings occur. Thus three points are shattered.

Four ordered points are not shattered: the alternating labeling $+,-,+,-$ has neither its positive set nor its negative set consecutive, so it cannot arise from an interval or its complement. Therefore