  A rainbow stacking of $r$-edge-colorings $\chi_1, \ldots, \chi_m$ of the
complete graph on $n$ vertices is a way of superimposing $\chi_1, \ldots,
\chi_m$ so that no edges of the same color are superimposed on each other. We
determine a sharp threshold for $r$ (as a function of $m$ and $n$) governing
the existence and nonexistence of rainbow stackings of random
$r$-edge-colorings $\chi_1,\ldots,\chi_m$.
