The k-set agreement problem is a generalization of the consensus problem. Namely, assuming that each process proposes a value, every non-faulty process should decide one of the proposed values, and no more than k different values should be decided. This is a hard problem in the sense that we cannot solve it in an asynchronous system, as soon as k or more processes may crash. One way to sidestep this impossibility result consists in weakening the termination property, requiring that a process must decide a value only if it executes alone during a long enough period of time. This is the well-known obstruction-freedom progress condition. Consider a system of n anonymous asynchronous processes that communicate through atomic read/write registers, and such that any number of them may crash. In this paper, we address and solve the challenging open problem of designing an obstruction-free k-set agreement algorithm using only (n − k + 1) atomic registers. From a shared memory cost point of view, our algorithm is the best algorithm known so far, thereby establishing a new upper bound on the number of registers needed to solve the problem, and in comparison to the previous upper bound, its gain is (n − k) registers. We then extend this algorithm into a space-optimal solution for the repeated version of k-set agreement, and an x-obstruction-free solution that employs (n − k + x) atomic registers (with 1 ≤ x ≤ k < n)