The overflow in the Katona Theorem

TL;DR

This paper investigates bounds on the size of families of subsets with union size constraints, sharpening classical results like the Katona Theorem for large n relative to r.

Contribution

It provides new tight bounds on the number of large subsets in families with union size restrictions, extending and sharpening the classical Katona Theorem.

Findings

Bound on the number of large subsets for n ≥ 6r.

Bound on the number of sets of size at least r for n > 3.5r.

Results are proven to be optimal and extend the classical 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 $n\geq 6r$ 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.