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In [[mathematics]], '''Milliken's tree theorem''' in [[combinatorics]] is a partition theorem generalizing [[Ramsey's theorem]] to infinite [[Tree (set theory)|trees]], objects with more structure than [[Set (mathematics)|sets]].


Let T be a finitely splitting rooted tree of height ω, n a positive integer, and <math>\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>\mathbb{S}^n_T=C_1 \cup ... \cup C_r</math> then for some strongly embedded infinite subtree R of T, <math>\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.  
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Define <math>\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>\mathbb{S}^n</math> [[partition regular]] for each n &lt; ω, but that the homogeneous subtree R guaranteed by the theorem is '''strongly embedded''' in T.  
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== Strong embedding ==
Call T an α-tree if each branch of T has cardinality α. Define Succ(p, P)= <math> \{ q \in P : q \geq p \}</math>, and <math>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>S \subset T</math>, and the partial order on S is induced from T,
* if <math>s \in S</math> is nonmaximal in S and <math>t \in IS(s,T)</math>, then <math>|Succ(t,T) \cap IS(s,S)|=1</math>,
* there exists a strictly increasing function from <math>\alpha</math> to <math>\beta</math>, such that <math>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==
 
#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.
 
[[Category:Ramsey theory]]
[[Category:Theorems in discrete mathematics]]
[[Category:Trees (set theory)]]

Latest revision as of 02:10, 24 November 2014


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