It is conjectured that the only way a failure detector (FD) can help solving n-process tasks is by providing k-set consensus for some \documentclass[12pt]{minimal}
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\begin{document}$${k\in\{1,\ldots,n\}}$$\end{document} among all the processes. It was recently shown by Zieliński that any FD that allows for solving a given n-process task that is unsolvable read-write wait-free, also solves (n − 1)-set consensus. In this paper, we provide a generalization of Zieliński’s result. We show that any FD that solves a colorless task that cannot be solved read-write k-resiliently, also solves k-set consensus. More generally, we show that every colorless task \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{T}}$$\end{document} can be characterized by its set consensus number: the largest \documentclass[12pt]{minimal}
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\begin{document}$${k\in\{1,\ldots,n\}}$$\end{document} such that \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{T}}$$\end{document} is solvable (k − 1)-resiliently. A task \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{T}}$$\end{document} with set consensus number k is, in the failure detector sense, equivalent to k-set consensus, i.e., a FD solves \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{T}}$$\end{document} if and only if it solves k-set consensus. As a corollary, we determine the weakest FD for solving k-set consensus in every environment, i.e., for all assumptions on when and where failures might occur.
