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

Two ordered points $x_1<x_2$ are shattered by intervals: an interval can contain neither, either one alone, or both. Three ordered points cannot be shattered, because an interval containing the first and third must also contain the middle one. Hence

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