Milliken's tree theorem
From Wikipedia, the free encyclopedia
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.
[edit] Strong Embedding
Call T an α-tree if each branch of T has cardinality α. Thus each α-tree has height α, but a tree with height α need not be an α-tree. Define Succ(p, 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:
, and the partial order on S is induced from T,- if
is nonmaximal in S and 
, then 
, - there exists a strictly increasing function from α to β, 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.
[edit] 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.
