  Given two graphs $G$ and $H$, we investigate for which functions $p=p(n)$ the
random graph $G_{n,p}$ (the binomial random graph on $n$ vertices with edge
probability $p$) satisfies with probability $1-o(1)$ that every
red-blue-coloring of its edges contains a red copy of $G$ or a blue copy of
$H$. We prove a general upper bound on the threshold for this property under
the assumption that the denser of the two graphs satisfies a certain
balancedness condition. Our result partially confirms a conjecture by the first
author and Kreuter, and together with earlier lower bound results establishes
the exact order of magnitude of the threshold for the case in which $G$ and $H$
are complete graphs of arbitrary size.
  In our proof we present an alternative to the so-called deletion method,
which was introduced by R\"odl and Ruci\'{n}ski in their study of symmetric
Ramsey properties of random graphs (i.e. the case $G=H$), and has been used in
many proofs of similar results since then.
