Rainbow Stackings of Random Edge-Colorings

A rainbow stacking of $r$-edge-colorings ${\chi }_1,\dots ,{\chi }_m$ of the complete graph on $n$ vertices is a way of superimposing ${\chi }_1,\dots ,{\chi }_m$ 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 ${\chi }_1,\dots ,{\chi }_m$.