 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-145/4/a/solution

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

Let

v_{i} \in F_{2}^{n}

be the incidence vector of

S_{i}

. Then

v_{i} \cdot v_{i} = ∣ S_{i} ∣ \equiv 1 (mod 2), v_{i} \cdot v_{j} = ∣ S_{i} \cap S_{j} ∣ \equiv 0 (mod 2)

for

i \neq = j

. If

\sum_{i} c_{i} v_{i} = 0

, taking the dot product with

v_{j}

gives

c_{j} = 0

. The vectors are linearly independent, so

m \leq n

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