On the theshold fo ainbow connection numbe $$$$ in andom gaphs

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) \({\mathrm {rc}}(G)\) of \(G\). We investigate sharp thresholds in the Erdős–Rényi random graph for the property “\({\mathrm {rc}}(G)\le r\)” 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 \(r \ge 3\), 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 \(r\).