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The possible correlation profiles of networks with a given scale-free degree distribution are restricted and bounded by maximally correlated configurations. Dissortative networks consist of nested bilayers, in which low-degree vertices are connected to high-degree vertices. The number of these bilayers attains a constant value for large network size . Assortative networks exhibit monolayers of low-degree vertices, the number of which grows monotonously with . Analytical relations for the Pearson correlation coefficient of these extremal configurations are derived and shown to provide lower and upper bounds on the possible values. Both bounds are found to vanish for large networks.