Milliken'S Tree Theorem

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What is Milliken's tree theorem?

In mathematics, Milliken's tree theorem in combinatorics is a partition theorem generalizing Ramsey's theorem to infinite trees, objects with more structure than sets. Let T be a finitely splitting rooted tree of height ω, n a positive integer, and the collection of all strongly embedded subtrees of T of height n. In one of its simple forms, Milliken's tree theorem states that if then for some strongly embedded infinite subtree R of T, for some i ≤ r. This immediately implies Ramsey's theorem; take the tree T to be a linear ordering on ω vertices. Define where T ranges over finitely splitting rooted trees of height ω. Milliken's tree theorem says that not only is partition regular for each n < ω, but that the homogeneous subtree R guaranteed by the theorem is strongly embedded in T.

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propositionramsey theorystatementtheoremtheorems in discrete mathematictheorytrees (set theory

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Milliken tree theorem

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