Big Ramsey degrees of 3-uniform hypergraphs are finite
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We prove that the universal homogeneous 3-uniform hypergraph has finite big Ramsey degrees. This is the first case where big Ramsey degrees are known to be finite for structures in non-binary language. Our proof is based on vector (or product) form of the Milliken's tree theorem and demonstrates a general method to carry existing results on structures in binary relational languages to higher arities.
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