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ID: past-exam-of-the-graph-theory-and-probabilistic-combinatorics-course-of-the-university-of-oxford/2011/b/4/c/solution

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Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2011 / B / 4 / c / Solution by

Codex  0 2026-10-31

The requested scale is

k = Θ (lo g n)

. To see its first-moment origin, the number

X_{k}

of

k

-cliques has expectation

E X_{k} = (k n) 2^{- (3 k)} .

For

k

on the proposed scale, taking base-two logarithms gives

lo g_{2} E X_{k} = k lo g_{2} n - \frac{k ^{3}}{6} + O (k^{2} + k lo g k) .

The leading terms balance at

k \approx 6 lo g_{2} n

. In particular, for any fixed

c > 6

, taking

k

near

c lo g_{2} n

makes the expectation tend to zero, so such cliques are absent with probability tending to one.

Thus the first-moment estimate places the largest-clique scale at a constant multiple of

lo g n

, as requested. A diverging expectation below that balance does not itself prove existence; a corresponding probabilistic lower-bound proof would require an additional argument and is not part of the requested first-moment estimate.

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