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What do the edge permuations in the unique games conjecture represent?

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These permutations represent the constraints on the labeling. for each edge, there is some set of labellings for the edge's vertices which is legal. Since we are talking about unique games, these legal labellings are limited in the following way: if (u,v) is an edge, then for any possible labeling of u, there exist a single legal labeling of v. This means that we can represent the legal labellings of any edge's vertices by a permutation $\phi :[k] \to [k]$ where $[k]$ is the set of labels.

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