The Unbalance of Set Systems

Abstract

The unbalance of an intersecting family ${\mathcal{F}}F$ is defined as $|{\mathcal{F}}| - d({\mathcal{F}})$, where $d({\mathcal{F}})$ is the maximum degree of ${\mathcal{F}}$ i.e. the maximum of $|\{F \in {\mathcal{F}} : x \in F\}|$ over all vertices x. We show that the unbalance of a k-uniform intersecting family is at most $\binom{n-3}{k-2}$ when n ≥ 6k ³ and we determine all families achieving this bound.