  We consider the class of exchangeable fragmentation-coagulation (EFC)
processes where coagulations are multiple and not simultaneous, as in a
$\Lambda$-coalescent, and fragmentation dislocates at finite rate an individual
block into sub-blocks of infinite size. We call these partition-valued
processes, simple EFC processes, and study the question whether such a process,
when started with infinitely many blocks, can visit partitions with a finite
number of blocks or not. When this occurs, one says that the process comes down
from infinity. We introduce two sharp parameters $\theta_{\star}\leq
\theta^{\star}\in [0,\infty]$, so that if $\theta^{\star}<1$, the process comes
down from infinity and if $\theta_\star>1$, then it stays infinite. We
illustrate our result with regularly varying coagulation and fragmentation
measures. In this case, the parameters $\theta_{\star},\theta^{\star}$ coincide
and are explicit.
