  The K{\L}R conjecture of Kohayakawa, {\L}uczak, and R\"odl is a statement
that allows one to prove that asymptotically almost surely all subgraphs of the
random graph G_{n,p}, for sufficiently large p : = p(n), satisfy an embedding
lemma which complements the sparse regularity lemma of Kohayakawa and R\"odl.
We prove a variant of this conjecture which is sufficient for most known
applications to random graphs. In particular, our result implies a number of
recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical
extremal combinatorial theorems. We also discuss several further applications.
