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 $$\mathbb{S}^n_T$$ 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 $$\mathbb{S}^n_T=C_1 \cup ... \cup C_r$$ then for some strongly embedded infinite subtree R of T, $$\mathbb{S}^n_R \subset C_i$$ for some i ≤ r.

This immediately implies Ramsey's theorem; take the tree T to be a linear ordering on ω vertices.

Define $$\mathbb{S}^n= \bigcup_T \mathbb{S}^n_T$$ where T ranges over finitely splitting rooted trees of height ω. Milliken's tree theorem says that not only is $$\mathbb{S}^n$$ partition regular for each n &lt; 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)= $$ \{ q \in P : q \geq p \}$$, and $$IS(p,P)$$ 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:


 * $$S \subset T$$, and the partial order on S is induced from T,
 * if $$s \in S$$ is nonmaximal in S and $$t \in IS(s,T)$$, then $$|Succ(t,T) \cap IS(s,S)|=1$$,
 * there exists a strictly increasing function from $$\alpha$$ to $$\beta$$, such that $$S(n) \subset T(f(n)).$$

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.