Arrow's celebrated impossibility theorem states that a sufficiently diverse domain of voter preference profiles cannot be mapped into social orders of the alternatives without violating at least one of three appealing conditions. Following Fishburn and Kelly, we define a set of strict preference profiles to be super Arrovian if Arrow's impossibility theorem holds for this set and each of its strict preference profile supersets. We write σ(m, n) for the size of the smallest super Arrovian set for m alternatives and n voters. We show that σ(m, 2) = [2m/m-2] and σ(3, 3) = 19. We also show that σ(m, n) is bounded by a constant for fixed n and bounded on both sides by a constant times 2n for fixed m. In particular, we find that limn→∞ σ(3, n)/2n = 3. Finally, we answer two questions posed by Fishburn and Kelly on the structure of minimum and minimal super Arrovian sets.