Ramsey theorem for trees with successor operation

TL;DR

This paper establishes a comprehensive Ramsey theorem for trees with successor operations, unifying several classical results and providing new proofs and applications in finite and infinite combinatorics.

Contribution

It introduces a general Ramsey theorem for successor trees, generalizing Carlson-Simpson and Milliken theorems, with applications to big Ramsey degrees and classical combinatorial theorems.

Findings

Provides a short proof of the Nešetřil-R"odl theorem

Recovers the Graham-Rothschild theorem

Offers a non-forcing proof of Zucker’s theorem on big Ramsey degrees

Abstract

We prove a general Ramsey theorem for trees with a successor operation. This theorem is a common generalization of the Carlson-Simpson Theorem and the Milliken Tree Theorem for regularly branching trees. Our theorem has a number of applications both in finite and infinite combinatorics. For example, we give a short proof of the unrestricted Ne\v{s}et\v{r}il-R\"odl theorem, and we recover the Graham-Rothschild theorem. Our original motivation came from the study of big Ramsey degrees - various trees used in the study can be viewed as trees with a successor operation. To illustrate this, we give a non-forcing proof of a theorem of Zucker on big Ramsey degrees.