Finite trees are Ramsey under topological embeddings

TL;DR

This paper proves that finite rooted binary plane trees form a Ramsey class under topological embeddings, extending Ramsey theory to a new class of relational structures with implications for combinatorics.

Contribution

It establishes that finite rooted binary plane trees are a Ramsey class under topological embeddings, a novel result in combinatorial and structural Ramsey theory.

Findings

Finite rooted binary plane trees form a Ramsey class.

The result applies to topological embeddings mapping leaves to leaves.

Implications for relational structures via rooted triple relation.

Abstract

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.