Self-Dual Ramsey Degrees for Trees

TL;DR

This paper investigates Ramsey properties for pairs of maps between trees, establishing the existence of Ramsey degrees and connecting these results to known theorems for linear orders.

Contribution

It introduces a characterization of Ramsey degrees for pairs of maps between trees and links these findings to existing linear order Ramsey theorems.

Findings

No Ramsey theorem for pairs of maps with colorings depending on both coordinates.

Characterization of Ramsey degrees for pairs of maps between trees.

Implication of tree results for linear order Ramsey theorems.

Abstract

We consider a Ramsey statement for pairs of maps between trees, where one is an embedding as defined by Deuber and the other is a rigid surjection as defined by Solecki. We show that there is no Ramsey Theorem for pairs of maps where the coloring depends on both coordinates. On the other hand, we give a characterization of the Ramsey degrees for such pairs. Furthermore, we show that our theorem on Ramsey Degrees for pairs of maps between trees implies the Ramsey Theorem for pairs of maps between linear orders as proved by Solecki.