On the threshold for rainbow connection number r in random graphs

We call an edge colouring of a graph G a rainbow colouring if every pair of
vertices is joined by a rainbow path, i.e., a path where no two edges have the
same colour. The minimum number of colours required for a rainbow colouring of
the edges of G is called the rainbow connection number (or rainbow
connectivity) rc(G) of G. We investigate sharp thresholds in the
Erd\H{o}s-R\'enyi random graph for the property "rc(G) <= r" where r is a fixed
integer. It is known that for r=2, rainbow connection number 2 and diameter 2
happen essentially at the same time in random graphs. For r >= 3, we conjecture
that this is not the case, propose an alternative threshold, and prove that
this is an upper bound for the threshold for rainbow connection number r.