Milliken's tree theorem explained

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

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 S ^{n} _{T = C} _{1} \cup ... \cup C _{r} then for some strongly embedded infinite subtree R of T, 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

S ^{n=} c up _{T} S ^{n} _{T} where T ranges over finitely splitting rooted trees of height ω. Milliken's tree theorem says that not only is S ^{n} 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)=

\{ 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.

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.