The Unbalance of Set Systems

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 ≥ 6k3 and we determine all families achieving this bound.