Ramsey theorem for trees with successor operation
Combinatorics
2023-11-14 v1 Discrete Mathematics
Logic
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.
Cite
@article{arxiv.2311.06872,
title = {Ramsey theorem for trees with successor operation},
author = {Martin Balko and David Chodounský and Natasha Dobrinen and Jan Hubička and Matěj Konečný and Jaroslav Nešetřil and Andy Zucker},
journal= {arXiv preprint arXiv:2311.06872},
year = {2023}
}
Comments
37 pages, 9 figures