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Programmable self-assembly is one of the most promising strategies for making ensembles of nanostructures from synthetic components. Yet, predicting the phase behavior that emerges from a complex mixture of many interacting species is difficult, and designing such a system to exhibit a prescribed behavior is even more challenging. In this article, I develop a mean-field model for predicting linker-mediated interactions between DNA-coated colloids, in which the interactions are encoded in DNA molecules dispersed in solution instead of in molecules grafted to particles’ surfaces. As I show, encoding interactions in the sequences of free DNA oligomers leads to new behavior, such as a re-entrant melting transition and a temperature-independent binding free energy per kBT. This unique phase behavior results from a per-bridge binding free energy that is a nonlinear function of the temperature and a nonmonotonic function of the linker concentration, owing to subtle entropic contributions. To facilitate the design of experiments, I also develop two scaling limits of the full model that can be used to select the DNA sequences and linker concentrations needed to program a specific behavior or favor the formation of a prescribed target structure. These results could ultimately enable the programming and tuning of hundreds of mutual interactions by designing cocktails of linker sequences, thus pushing the field toward the original goal of programmable self-assembly: these user-prescribed structures can be assembled from complex mixtures of building blocks through the rational design of their interactions.