The overflow in the Katona Theorem

Let \(n›2r›0\) be integers. We consider families \(F\) of subsets of an \(n\)-element set, in which the union of any two members has size at most \(2r\). One of our results states that for \(n≥ 8r\) the number of members of size exceeding \(r\) in \(F\) is at most \({n-2}{r-1}\). Another result shows that for \(n›3.5r\) the number of sets of size at least \(r\) is at most \({n}{r}\). Both bounds are best possible and the latter sharpens the classical Katona Theorem. Similar results are proved for the odd case of the Katona Theorem as well. Mathematics Subject Classifications: 05D05 Keywords: The Katona theorem, overflow, shifting, the random walk method