  Kang, Perkins and Spencer showed that the size of the largest component of
the Bohman-Frieze process at a fixed time $t$ smaller than $t_c$, the critical
time for the process is $L_1(t)=\Omega(\log n/(t_c-t)^2)$ with high
probability. They also conjectured that this is the correct order, that is
$L_1(t)=O(\log n/(t_c-t)^2)$ with high probability for fixed $t$ smaller than
$t_c$. Using a different approach, Bhamidi, Budhiraja and Wang showed that
$L_1(t_n)=O((\log n)^4/(t_c-t_n)^2)$ with high probability for $t_n\leq
t_c-n^{-\gamma}$ where $\gamma\in(0,1/4)$. In this paper, we improve their
result by showing that for any fixed $\lambda>0$, $L_1(t_n)=O(\log
n/(t_c-t_n)^2)$ with high probability for $t_n\leq t_c-\lambda n^{-1/3}$. In
particular, this settles the conjecture of Kang, Perkins and Spencer. We also
prove some generalizations for general bounded size rules.
