We show that the class of finite rooted binary plane trees is a Ramsey class (with respect to topological embeddings that map leaves to leaves). That is, for all such trees P,H and every natural number k there exists a tree T such that for every k-coloring of the (topological) copies of P in T there exists a (topological) copy H' of H in T such that all copies of P in H' have the same color. When the trees are represented by the so-called rooted triple relation, the result gives rise to a Ramsey class of relational structures with respect to induced substructures.
As we have mentioned in the previous subsection, any Ramsey class that is closed under taking substructures (and our Ramsey class @math is obviously closed under taking substructures) is an amalgamation class, and therefore there exists a unique countable homogeneous structure @math such that @math is exactly the class of all finite structures that embed into @math . The structure @math (i.e., the reduct of @math that only contains the rooted triple relation without an ordering on the domain) is well-known to model-theorists and in the theory of infinite permutation groups, and also has many explicit constructions; see e.g. @cite_4 @cite_5 . Its automorphism group is oligomorphic, 2-transitive, and 3-set-transitive, but not 3-transitive. The rooted triple relation @math is a @math -relation in the terminology of @cite_5 .
@cite_5 @cite_4
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Preparation Semilinear order relations Abstract chain sets General betweenness relations Abstract direction sets Applications and commentary References. 1. Introduction 2. Preliminaries 3. Examples and growth rates 4. Subgroups 5. Miscellaneous topics.