A rainbow stacking of $r$-edge-colorings $χ_1, \ldots, χ_m$ of the complete graph on $n$ vertices is a way of superimposing $χ_1, \ldots, χ_m$ 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 $χ_1,\ldots,χ_m$.
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