A Dual Ramsey Theorem for Permutations

Abstract

In 2012 M. Sokić 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. Böttcher and J. Foniok, in 2014 by M. Bodirsky and in 2015 yet another proof was provided by M. Sokić.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.

A dual Ramsey theorem for finite ordered oriented graphs

“…We shall also need a categorical version of the Product Ramsey Theorem for Finite Structures of M. Sokić [18]. We proved this statement in the categorical context in [7] where we used this abstract version to prove that the class of finite permutations has the dual Ramsey property.…”

“…Our goal in this paper is to prove that the category OOgra srq has the dual Ramsey property. In order to do so, we shall employ a strategy devised in [7]. Let us recall two technical statements from [7].…”

“…In order to do so, we shall employ a strategy devised in [7]. Let us recall two technical statements from [7].…”

“…Theorem 4.1 [7] Let C be a category such that every morphism in C is monic and such that hom C (A, B) is finite for all A, B ∈ Ob(C), and let D be a (not necessarily full) subcategory of C. If C has the Ramsey property and D is closed for binary diagrams, then D has the Ramsey property.…”

“…Theorem 4.2 [7] Let C 1 and C 2 be categories such that hom C i (A, B) is finite for all A, B ∈ Ob(C i ), i ∈ {1, 2}. If C 1 and C 2 both have the Ramsey property then C 1 × C 2 has the Ramsey property.…”