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.

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. Call T an α-tree if each branch of T has cardinality α.

to be the set of immediate successors of p in P. Suppose S is an α-tree and T is a β-tree, with 0 ≤ α ≤ β ≤ ω.