# A Dual Ramsey Theorem for Permutations

## TL;DR

This paper extends the understanding of permutation classes by proving they possess the dual Ramsey property using categorical methods, building on prior results about their original Ramsey property.

## Contribution

It introduces a categorical approach to establish the dual Ramsey property for all finite permutations, complementing existing proofs of the original property.

## Key findings

- Finite permutations have the dual Ramsey property.
- Categorical methods effectively prove dual properties.
- Supports the broader applicability of category theory in structural Ramsey theory.

## Abstract

In 2012 M. Soki\'c proved that the class of all finite permutations has the Ramsey property. Using different strategies the same result was then reproved in 2013 by J. B\"ottcher and J. Foniok, in 2014 by M. Bodirsky and in 2015 yet another proof was provided by M. Soki\'c. Using the categorical reinterpretation of the Ramsey property in this paper we prove that the class of all finite permutations has the dual Ramsey property as well. It was Leeb who pointed out in 1970 that the use of category theory can be quite helpful both in the formulation and in the proofs of results pertaining to structural Ramsey theory. In this paper we argue that this is even more the case when dealing with the dual Ramsey property.