 Solution

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

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

For one real coordinate, a union of

m

intervals labels a fixed

n

-point sample by choosing at most

2 m

endpoints among the

n + 1

gaps, so the number of labelings is at most

(n + 1)^{2 m}

. Membership in a Cartesian product is the product of the

p

coordinate-membership indicators. Applying part (c) repeatedly gives

s (F_{3}, n) \leq ((n + 1)^{2 m})^{p} = (n + 1)^{2 m p} .

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