The overflow in the Katona Theorem

Let \(n›2r›0\) be integers. We consider families \(\mathcal{F}\) 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 \(n\geq 8r\) the number of members of size exceeding \(r\) in \(\mathcal{F}\) is at most \(\binom{n-2}{r-1}\). Another result shows that for \(n›3.5r\) the number of sets of size at least \(r\) is at most \(\binom{n}{r}\). 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