We study the computational complexity of problems that ask if a given graph admits an edge-coloring that does not contain an edge-colored clique from some fixed finite family. We show that every such problem is poly-time equivalent to a Constraint Satisfaction Problem, yielding a P vs. NP-complete dichotomy. Our main contribution lies in the reduction from the CSP to the coloring problem where we apply methods from Ramsey theory and a novel notion of cut-homotopy.

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