K\H{o}nig = Ramsey, A compactness lemma for Ramsey categories
Maximilian Hadek
TL;DR
This paper introduces a new characterization of the Ramsey property in categories using a generalized Kőnig's lemma, leading to novel proofs and transfer theorems in structural Ramsey theory.
Contribution
It provides a novel categorical characterization of the Ramsey property and unifies various Ramsey transfer results through a new transfer theorem involving Grothendieck opfibrations.
Findings
New proof of existence and uniqueness of minimal Ramsey expansions
A transfer theorem unifying several known Ramsey transfers
Application of a generalized Kőnig's lemma to Ramsey categories
Abstract
We prove a new characterization of the Ramsey property of categories in terms of a generalized form of K\H{o}nig's tree lemma. Afterwards, we discuss its applications to structural Ramsey theory. In particular, we provide a new proof of the existence and uniqueness of minimal Ramsey expansions and a new transfer theorem which uses Grothendieck opfibrations and unifies several known Ramsey transfers.
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