  Bounded-size rules are dynamic random graph processes which incorporate
limited choice along with randomness in the evolution of the system. One starts
with the empty graph and at each stage two edges are chosen uniformly at
random. One of the two edges is then placed into the system according to a
decision rule based on the sizes of the components containing the four
vertices. For bounded-size rules, all components of size greater than some
fixed $K\geq 1$ are accorded the same treatment. Writing $\BS(t)$ for the state
of the system with nt/2 edges, Spencer and Wormald proved that for such rules,
there exists a critical time t_c such that when t< t_c the size of the largest
component is of order $\log{n}$ while for $t> t_c$, the size of the largest
component is of order $n$. In this work we obtain upper bounds (that hold with
high probability) of order $n^{2\gamma} \log ^4 n$, on the size of the largest
component, at time instants $t_n = t_c-n^{-\gamma}$, where $\gamma\in (0,1/4)$.
This result for the barely subcritical regime forms a key ingredient in the
study undertaken in \cite{amc-2012}, of the asymptotic dynamic behavior of the
process describing the vector of component sizes and associated complexity of
the components for such random graph models in the critical scaling window. The
proof uses a coupling of BSR processes with a certain family of inhomogeneous
random graphs with vertices in the type space $\Rbold_+\times
\cD([0,\infty):\NNN_0)$ where $\cD([0,\infty):\NNN_0)$ is the Skorohod
$D$-space of functions that are right continuous and have left limits equipped
with the usual Skorohod topology. The coupling construction also gives an
alternative characterization (than the usual explosion time of the
susceptibility function) of the critical time $t_c$ for the emergence of the
giant component in terms of the operator norm of integral operators on certain
$L^2$ spaces.
