Second moment method
Hints: 25 November 2015
Deadline: 2 December 2015
1. Let Ai , Bi be finite sets (i = 1, . . . , m) such that Ai ∩ Bi = ∅ for every i, but at least one of
Ai ∩ Bj , Aj ∩ Bi is nonempty for every i 6= j. Show that for every p ∈ [0, 1] it holds that
m
X
p|Ai | (1 − p)|Bi | ≤ 1.
i=1
[2 points]
Moreover, if |Ai | = a, |Bi | = b for every i, prove that
m≤
(a + b)(a+b)
.
a a bb
[1 point]
2. Let X be a random variable with Var[X] = σ 2 and E[X] = 0. For every real λ > 0 prove
the inequality
σ2
Pr[X ≥ λ] ≤ 2
.
σ + λ2
[2 points]
3. Let X be a random variable taking only nonnegative integer values. Moreover, E[X 2 ] is
finite and nonzero. Show that
Var[X]
.
Pr[X = 0] ≤
E[X 2 ]
[2 points]
4. Let v1 = (x1 , y1 ), . . . , vn = √
(xn , yn ) be n two-dimensional vectors such that xi , yi are integers
with |xi |, |yi | ≤ 2n/2 /(100 n). Show that there exist two nonempty disjoint sets I, J ⊆
{1, . . . , n} such that
X
X
vi =
vj .
i∈I
j∈J
[4 points]
5. Prove that there exists a real constant c > 0 P
such that the following holds: for every integer
n
n and for every a1 , a2 , . . . , an ∈ R satisfying i=1 a2i = 1, if we for every i (1 ≤ i ≤ n) take
εi ∈ {−1, 1} uniformly independently at random, then
X
n
Pr εi ai ≤ 1 ≥ c.
i=1
[4 points]
6. Prove that for every sufficiently large n there exists a bipartite graph G = (L, R, E) satisfying
the following:
• |L| = |R| = n,
• every vertex in L has degree at most 18, and
• every subset S ⊆ L with |S| ≤ n/3 has at least 2|S| neighbors in R.
[5 points]
1