The overflow in the Katona Theorem

Abstract

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 $n\ge 6r$ the number of members of size exceeding $r$ in $\mathcal{F}$ is at most $\left(\genfrac{}{}{0ex}{}{n-2}{r-1}\right)$. Another result shows that for $n>3.5r$ the number of sets of size at least $r$ is at most $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$. 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.