A stability result for the Katona theorem

Let n, s be positive integers, n≥s+2. In 1964 Katona [5] established the maximum possible size of a family of subsets of {1,2,…,n} such that the union of any two members of the family has size of at most s. Katona also proved that the optimal families are unique up to isomorphism. In the present paper we sharpen this result by showing that excluding those optimal families one can get better bounds. These new bounds are best possible.