= Compactness extension of the Lovász local lemma
For countably many constraints on finite-valued variables, each involving finitely many variables, apply the finite <Lovász local lemma> to every finite constraint subfamily. Partial satisfying assignments on increasing finite variable sets form a finitely branching <tree> with nonempty levels. The <König infinity lemma> gives an infinite branch, producing one assignment satisfying all constraints. This existence argument does not assert positive probability of simultaneous avoidance in an infinite product space.
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