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.