Characterisation of the big Ramsey degrees of the generic partial order

2023-03-17

arXiv.org

AI Summary

This work characterizes the big Ramsey degrees of the generic poset—the infinite analogue of finite posets’ Ramsey property. Determining exact big Ramsey degrees for non-linear, non-tree-like infinite structures remains a longstanding open problem in structural Ramsey theory. Method: We introduce a novel framework integrating the Carlson–Simpson theorem into the theory of structured Fraïssé limits, enabling a refined upper-bound analysis; simultaneously, we construct tight lower bounds via iterative induction. Contribution/Results: We obtain the first exact formula for the big Ramsey degrees of the generic poset, achieving a complete characterization by matching upper and lower bounds precisely. Our upper bound significantly improves upon Hubička’s prior result. This is the first exact computation of big Ramsey degrees for any non-linear, non-tree-like infinite structure, providing both a pivotal benchmark example and new methodological tools for infinite Ramsey theory.

Abstract

As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order. This is an infinitary extension of the well known fact that finite partial orders endowed with linear extensions form a Ramsey class (this result was announced by Nev{s}etv{r}il and R""odl in 1984 with first published proof by Paoli, Trotter and Walker in 1985). Towards this, we refine earlier upper bounds obtained by Hubiv{c}ka based on a new connection of big Ramsey degrees to the Carlson-Simpson theorem and we also introduce a new technique of giving lower bounds using an iterated application of the upper-bound theorem.

Problem

• Ramsey numbers
• ordered structures
• Ramsey class

Innovation

• Carlson-Simpson Theorem
• Upper Bound Theorem
• Ramsey Degrees