Milliken's tree theorem

Short description: Theorem in combinatorics generalizing Ramsey's theorem to infinite trees

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

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

Define [math]\displaystyle{ \mathbb{S}^n= \bigcup_T \mathbb{S}^n_T }[/math] where T ranges over finitely splitting rooted trees of height ω. Milliken's tree theorem says that not only is [math]\displaystyle{ \mathbb{S}^n }[/math] 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)= [math]\displaystyle{ \{ q \in P : q \geq p \} }[/math], and [math]\displaystyle{ IS(p,P) }[/math] 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:

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

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

1. Keith R. Milliken, A Ramsey Theorem for Trees J. Comb. Theory (Series A) 26 (1979), 215-237
2. Keith R. Milliken, A Partition Theorem for the Infinite Subtrees of a Tree, Trans. Amer. Math. Soc. 263 No.1 (1981), 137-148.