 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/31j/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Choose

2 m

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

2 m + 1

ordered points, however, the alternating labeling

1, 0, 1, 0, \dots, 0, 1

has

m + 1

separated positive runs and cannot be realized by a union of

m

intervals. Therefore

VC (F_{1}) = 2 m .

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