Several Ramsey theorems and Ramsey spaces, such as the Milliken tree theorem and the Carlson-Simpson theorem, are naturally viewed as results about trees and their subtrees. Recently, the study of big Ramsey degrees of universal structures has led to a need for additional variants of these theorems where the notion of a subtree is modified.

We discuss a general framework for proving Ramsey-type theorems on trees with finite but possibly unbounded branching and the associated Ramsey spaces. These spaces are formed by collections of infinite subtrees equipped with a topology generalizing the Ellentuck space. By verifying that these structures satisfy the abstract Ramsey space axioms, we ensure that every subset with the Baire property is Ramsey. This framework specifically incorporates the successor operation to maintain structural integrity during embeddings.

This is joint work with Martin Balko, David Chodounský, Natasha Dobrinen, Matěj Konečný, Jaroslav Nešetřil, and Andy Zucker.