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You may find the multiplicative Chernoff bound useful: if X ∼ Bin(m, p) then for any 0 < δ < 1,
Pr[(1 − δ)pm ≤ X ≤ (1 + δ)pm] ≥ 1 − 2e^(−(δ^2)*pm/3)
We say that a graph is α-dense if its density is at least α.
Show that for each constant 1/2 < α ≤ 1, whp the largest α-dense subgraph of G(n, 1/2) has size Θα(log n).
(This means that there are constants Aα, Bα > 0 such that whp, the size is between Aα log n and Bα log n.)
Sagot :
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