  Let $G$ be a group and $H_1$,...,$H_s$ be subgroups of $G$ of indices
$d_1$,...,$d_s$ respectively. In 1974, M. Herzog and J. Sch\"onheim conjectured
that if $\{H_i\alpha_i\}_{i=1}^{i=s}$, $\alpha_i\in G$, is a coset partition of
$G$, then $d_1$,..,$d_s$ cannot be distinct. We consider the Herzog-Sch\"onheim
conjecture for free groups of finite rank and develop a new combinatorial
approach, using covering spaces. We define $Y$ the space of coset partitions of
$F_n$ and show $Y$ is a metric space with interesting properties. We give some
sufficient conditions on the coset partition that ensure the conjecture is
satisfied and moreover has a neighborhood $U$ in $Y$ such that all the
partitions in $U$ satisfy also the conjecture.
