  We establish a first general partial regularity theorem for area minimizing
currents $\mathrm{mod}(p)$, for every $p$, in any dimension and codimension.
More precisely, we prove that the Hausdorff dimension of the interior singular
set of an $m$-dimensional area minimizing current $\mathrm{mod}(p)$ cannot be
larger than $m-1$. Additionally, we show that, when $p$ is odd, the interior
singular set is $(m-1)$-rectifiable with locally finite $(m-1)$-dimensional
measure.
