Chapter 4 Non-uniform Hypergraphs
4.4 Improving Theorem 4.1
In Theorem 4.1, q(H,p) ≥ > 0 was a sufficient condition for obtaining copies of
H inGR(n,p). It turns out that we can invoke Chebyshev’s inequality to lower this threshold to o(1). Let ω(n) denote a function that tends slowly to ∞ as n→ ∞.
Theorem 4.4. If q(H,p) ≥ loglogω(nn), then almost surely GR(n,p) contains (1 +
o(1))|Aut1(H)|nφ(H,p) copies of H.
Theorem 4.4 follows from the following lemma.
Lemma 4.5. If E(X) = Ω(ω(n)), then Pr(|X−E(X)| ≥λ) = o(1), where λ = maxn1 nE(X), q E(X)ω(n) o . Proof. Observe that
X S Var (XS) = X S (E(XS2)−(E(XS))2) ≤ X S E(XS2) = X S E(XS) = E(X).
If|S∩S0| ≤1, then XS and XS0 are independent, in which case Covar(XS, XS0) = 0. Hence, X S6=S0 Covar(XS, XS0) ≤ X S6=S0 |S∩S0|≥2 |Covar(XS, XS0)| = X S6=S0 |S∩S0|≥2 |E(XSXS0)−E(XS)E(XS0)| ≤ X S E(XS) X S06=S |S∩S0|≥2 |E(XS0|XS)−E(XS0)| ≤ X S E(XS) X S06=S E(XS0|XS) + X S E(XS) X S06=S |S∩S0|≥2 E(XS0) ≤ X S E(XS) X H0 n −Pr i=1αifi(H 0) + n −Pr i=1αifi(H) ≤ X S E(XS) nq(H,p)+ 1 n2E(X) ≤ E(X)nq(H,p)+ 1 n2(E(X)) 2.
Putting these together,
Var(X) = Var X S XS ! (4.1) = X S Var (XS) + X S6=S0 Covar(XS, XS0) (4.2) ≤ E(X)(1 +nq(H,p)) + 1 n2(E(X)) 2 . (4.3)
Now Chebyshev’s inequality states that forλ >0, Pr (|X−E(X)| ≥λ)≤ Var(X)
λ2 . (4.4)
Choose λ = maxnn1E(X),qE(X)ω(n)o. Thus λ =o(E(X)). Hence, applying (4.3) to (4.4) implies Pr (|X−E(X)| ≥λ) ≤ max ( 1 n2, 1 ω(n) ) = o(1).
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