Rainbow copies of spanning subgraphs

Let $G_{n,p}^{[\kappa ]}$ denote the space of $n$-vertex edge coloured graphs, where each edge occurs independently with probability $p$. The colour of each existing edge is chosen independently and uniformly at random from the set $[\kappa ]$. We consider the threshold for the existence of rainbow colored copies of a spanning subgraph $H$. We provide lower bounds on $p$ and $\kappa$ sufficient to prove the existence of such copies w.h.p.