This paper considers the eventual leader election problem in asynchronous message-passing systems where an arbitrary number t of processes can crash (t < n, where n is the total number of processes). It considers weak assumptions both on the initial knowledge of the processes and on the network behavior. More precisely, initially, a process knows only its identity and the fact that the process identities are different and totally ordered (it knows neither n nor t). Two eventual leader election protocols are presented. The first protocol assumes that a process also knows the lower bound α on the number of processes that do not crash. This protocol requires the following behavioral properties from the underlying network: the graph made up of the correct processes and fair lossy links is strongly connected, and there is a correct process connected to t − f other correct processes (where f is the actual number of crashes in the considered run) through eventually timely paths (paths...