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. Strong embedding Call T an α-tree if each branch of T has cardinality α. Define Succ(p, P)= , and 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. S is strongly embedded in T if: , and the partial order on S is induced from T, if is nonmaximal in S and , then , there exists a strictly increasing function from to , such that Intuitively, for S to be strongly embedded in T, S must be a subset of T with the induced partial order S must preserve the branching structure of T; i.e., if a nonmaximal node in S has n immediate successors in T, then it has n immediate successors in S S preserves the level structure of T; all nodes on a common level of S must be on a common level in T. References Keith R. Milliken, A Ramsey Theorem for Trees J. Comb. Theory (Series A) 26 (1979), 215-237 Keith R. Milliken, A Partition Theorem for the Infinite Subtrees of a Tree, Trans. Amer. Math. Soc. 263 No.1 (1981), 137-148. Ramsey theory Theorems in discrete mathematics Trees (set theory)