 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-2/30j/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

For each

d

-element coordinate set

J \subseteq {1, \dots, p}

, let

G_{J}

contain the lower orthants whose finite thresholds occur precisely in the coordinates in

J

. Ignoring the remaining coordinates identifies

G_{J}

with the lower-orthant class in

R^{d}

, so part (a) gives

VC (G_{J}) = d

. Moreover,

G = ⋃_{∣ J ∣ = d} G_{J}

is a union of

r = (d p)

such classes. Part (c) and

(d p) \leq p^{d}

give

VC (G) \leq 4 d lo g_{2} (2 d) + 2 lo g_{2} (d p) \leq 4 d lo g_{2} (2 d) + 2 d lo g_{2} p = 2 d {2 lo g_{2} (2 d) + lo g_{2} p} .

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