The complexity of identifying large equivalence classes

We prove that at least 3k−4/k(2k−3)(n/2) - O(k)equivalence tests and no more than 2/k (n/2) + O(n) equivalence tests are needed in the worst case to identify the equivalence classes with at least k members in set of n elements. The upper bound is an improvement by a factor 2 compared to known results. For k = 3 we give tighter bounds. Finally, for k n/2 we prove that it is necessary and it suffices to make 2n − k − 1 equivalence tests which generalizes a known result for k = [n+1/2].