The recent discovery of the weakest failure detector for message passing set agreement has renewed the interest in exploring the border between solvable and unsolvable problems in message passing systems. This paper contributes to this research by introducing two novel system models Manti and M sink with very weak synchrony requirements, where L can be implemented. To the best of our knowledge, they are the first message passing models where set agreement is solvable but consensus is not. We also generalize L by a novel "(n-k)-loneliness" failure detector L(k), which allows to solve k-set agreement but not (k-1)-set agreement. We also present an algorithm that solves k-set agreement with L(k), which is anonymous in that it does not require unique process identifiers. This reveals that L is also the weakest failure detector for anonymous set agreement. Finally, we analyze the relationship between L(k) and other failure detectors, namely the limited scope failure detector Sn-k+1 and the quorum failure detector ∑.