In their seminal work which initiated random graph theory Erdös and Rényi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let
V
n
(
p
)
=
{
0
,
1
}
n
V_n(p)= \{0,1\}^n
denote the Hamming space endowed with the probability measure
μ
p
\mu _p
defined by
μ
p
(
ϵ
1
,
ϵ
2
,
…
,
ϵ
n
)
=
p
k
⋅
(
1
−
p
)
n
−
k
\mu _p (\epsilon _1, \epsilon _2, \dots , \epsilon _n)= p^k \cdot (1-p)^{n-k}
, where
k
=
ϵ
1
+
ϵ
2
+
⋯
+
ϵ
n
k=\epsilon _1 +\epsilon _2 +\cdots +\epsilon _n
. Let
A
A
be a monotone subset of
V
n
V_n
. We say that
A
A
is symmetric if there is a transitive permutation group
Γ
\Gamma
on
{
1
,
2
,
…
,
n
}
\{1,2,\dots , n\}
such that
A
A
is invariant under
Γ
\Gamma
.
Theorem. For every symmetric monotone
A
A
, if
μ
p
(
A
)
>
ϵ
\mu _p(A)>\epsilon
then
μ
q
(
A
)
>
1
−
ϵ
\mu _q(A)>1-\epsilon
for
q
=
p
+
c
1
log
(
1
/
2
ϵ
)
/
log
n
q=p+ c_1 \log (1/2\epsilon )/\log n
. (
c
1
c_1
is an absolute constant.)