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Let $\\mathcal{D}(n,m)$ be a graph with $m$ edges obtained after $m$ steps of this process. Each edge $e_i$ ($i=1,2,\\ldots, m$) of $\\mathcal{D}(n,m)$ independently chooses a colour, taken uniformly at random from a given set of $n(1 + O( \\log \\log n / \\log n)) = n (1+o(1))$ colours. 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