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ID: past-exam-of-the-graph-theory-and-probabilistic-combinatorics-course-of-the-university-of-oxford/2009/a/3/d/solution

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Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2009 / A / 3 / d / Solution by

Codex  0 2026-10-31

Apply part (c) to the pairs

(X_{1}, X_{2})

and

(X_{1}, X_{3})

. Outside a union of at most

2 ϵ n

exceptional vertices, every

v \in X_{1}

has sets

Y = N (v) \cap X_{2}

and

Z = N (v) \cap X_{3}

of size at least

d n /2

.

These sets are large enough for epsilon-uniformity of

(X_{2}, X_{3})

, because

d /2 > ϵ

. Hence

e (Y, Z) \geq (d - ϵ) ∣ Y ∣∣ Z ∣ \geq \frac{d}{2} (\frac{d n}{2})^{2} = \frac{d ^{3} n ^{2}}{8} .

Every such edge completes a distinct triangle through

v

. Thus at most

2 ϵ n

X_{1}

fail the claimed triangle count. Summing these triangles over the good vertices counts each triangle with one vertex in each class exactly once. Since

d \leq 1

and

ϵ < d /4 \leq 1/4

,

# {triangles in G} \geq (1 - 2 ϵ) n \frac{d ^{3} n ^{2}}{8} \geq \frac{d ^{3} n ^{3}}{16} .

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