A \emph{failure detector} is a distributed oracle that provides the processes with information about failures. The \emph{perfect} failure detector provides accurate and eventually complete information about process failures. We show that, in asynchronous failure-prone message-passing systems, perfect failure detection can be achieved by an oracle that outputs at most $\lceil \log \alpha(n)\rceil+1$ bits per process in $n$-process systems, where $\alpha$ denotes the inverse-Ackermann function. This result is essentially optimal, as we also show that, in the same environment, no failure detectors outputting a constant number of bit per process can achieve perfect failure detection.