Isomorphy up to complementation

Considering uniform hypergraphs, we prove that for every non-negative integer h there exist two non-negative integers k and t with k ≤ t such that two h-uniform hypergraphs H and H′ on the same set V of vertices, with ∣V ∣ ≥ t, are equal up to complementation whenever H and H′ are k-hypomorphic up to complementation. Let s(h) be the least integer k such that the conclusion above holds and let v(h) be the least t corresponding to k = s(h). We prove that s(h) = h + 2⌊log2 h⌋. In the special case h = 2 or h = 2 + 1, we prove that v(h) ≤ s(h) + h. The values s(2) = 4 and v(2) = 6 were obtained in [9].