Sperner systems containing at most k sets of every cardinality

We prove using a direct construction that one can choose n − 2 subsets of an n-element set with different cardinality such that none of them contains any other. As a generalization, we prove that if for any j we can have at most k subsets containing exactly j elements (k> 1), then for n 5 we can choose at most k(n − 3) subsets from an n-element set such that they form a Sperner system. Moreover, we prove that this can be achieved if n is large enough, and give a construction for n 8k − 4.