  Let $k$ be a positive integer and $G$ be a finite group that cannot be
written as the union of $k$ proper subgroups. In this short note, we study the
existence of a constant $c_k\in (0,1)$ such that $|\cup_{i=1}^k H_i| \leq
c_k|G|$, for all proper subgroups $H_1$, ..., $H_k$ of $G$.
