Categorical view of the Partite Lemma in structural Ramsey Theory

Abstract

We construct of the main object of the Partite Lemma as the colimit over a certain diagram. This gives a purely category theoretic take on the Partite Lemma and establishes the canonicity of the object. Additionally, the categorical point of view allows us to unify the direct Partite Lemma in [4], [5], and [6] with the dual Paritite Lemma in [8].

Finite Ramsey Theory through Category Theory

“…The importance of finiteness of Ramsey degree stems partly from it being a refinement of the Ramsey property and partly, and more significantly, from its relevance to topological dynamics as indicated in [6] and, especially, in [15]. Going beyond just formulating Ramsey theoretic notions, several papers used the language of category theory to carry out proofs of finite Ramsey theoretic statements; see, for example, [2], [5], [7], [8], and [9]. This was usually done by identifying a category A that was known to be Ramsey or to have a related property, and then transferring Ramseyness or the related property from A to another category B by finding an, often subtle, connection between A and B.…”

“…In a number of specific categories, he verified certain identities, called by him Pascal identities, and then working separately in each of these categories, but using similarly structured arguments, he showed that the categories are Ramsey, from which various classical Ramsey theorems followed. In [5], certain Ramsey theoretic constructions were revealed to be canonical category theory constructions. In his paper [3], Gromov advocated for a broad use of category theory in Ramsey theory.…”