We study leader election in unidirectional rings of homonyms that have no a priori knowledge of the number of processes. In this context, we show that there exists no algorithm that solves the process-terminating leader election problem for the class of asymmetrically labeled unidirectional rings. More precisely, we prove that there is no process-terminating leader election algorithm even for the subclass of unidirectional rings where at least one label is unique. Message-terminating leader election is also impossible for the class of unidirectional rings where only a bound on multiplicity is known. However, we show that the processterminating leader election is possible for two particular subclasses of asymmetrically labeled unidirectional rings where the multiplicity is bounded. We propose three efficient algorithms and analyze their complexities. We also give some non-trivial lower bounds.