Probabilistic methods have been used to approach many problems of Ramsey theory. In this paper we study Ramsey type questions from the point of view of random structures. Let
K
(
n
,
N
)
K(n,N)
be the random graph chosen uniformly from among all graphs with
n
n
vertices and
N
N
edges. For a fixed graph
G
G
and an integer
r
r
we address the question what is the minimum
N
=
N
(
G
,
r
,
n
)
N = N(G,r,n)
such that the random graph
K
(
n
,
N
)
K(n,N)
contains, almost surely, a monochromatic copy of
G
G
in every
r
r
-coloring of its edges (
K
(
n
,
N
)
→
(
G
)
r
K(n,N) \to {(G)_r}
, in short). We find a graph parameter
γ
=
γ
(
G
)
\gamma = \gamma (G)
yielding
\[
lim
n
→
∞
Prob
(
K
(
n
,
N
)
→
(
G
)
r
)
=
{
0
if
N
>
c
n
y
,
1
if
N
>
C
n
y
,
\lim \limits _{n \to \infty } \operatorname {Prob}(K(n,N) \to {(G)_r}) = \left \{ {\begin {array}{*{20}{c}} {0\quad {\text {if }}\;N > c{n^y},} \\ {1\quad {\text {if}}\;N > C{n^y},} \\ \end {array} } \right .\quad
\]
for some
c
c
,
C
>
0
C > 0
. We use this to derive a number of consequences that deal with the existence of sparse Ramsey graphs. For example we show that for all
r
≥
2
r \geq 2
and
k
≥
3
k \geq 3
there exists
C
>
0
C > 0
such that almost all graphs
H
H
with
n
n
vertices and
C
n
2
k
k
+
1
C{n^{\frac {{2k}}{{k + 1}}}}
edges which are
K
k
+
1
{K_{k + 1}}
-free, satisfy
H
→
(
K
k
)
r
H \to {({K_k})_r}
. We also apply our method to the problem of finding the smallest
N
=
N
(
k
,
r
,
n
)
N = N(k,r,n)
guaranteeing that almost all sequences
1
≤
a
1
>
a
2
>
⋯
>
a
N
≤
n
1 \leq {a_1} > {a_2} > \cdots > {a_N} \leq n
contain an arithmetic progression of length
k
k
in every
r
r
-coloring, and show that
N
=
Θ
(
n
k
−
2
k
−
1
)
N = \Theta ({n^{\frac {{k - 2}}{{k - 1}}}})
is the threshold.