  We study the intersecting family process initially studied in \cite{BCFMR}.
Here $k=k(n)$ and $E_1,E_2,\ldots,E_m$ is a random sequence of $k$-sets from
$\binom{[n]}{k}$ where $E_{r+1}$ is uniformly chosen from those $k$-sets that
are not already chosen and that meet $E_i,i=1,2,\ldots,r$. We prove some new
results for the case where $k=cn^{1/3}$ and for the case where $k\gg n^{1/2}$.
