A new perspective on semi-retractions and the Ramsey property

TL;DR

This paper explores semi-retractions between structures, linking the Ramsey property and indiscernible sequences, and establishes conditions for transferring Ramsey properties, enhancing the understanding of combinatorics and model theory connections.

Contribution

It introduces a new perspective on semi-retractions, demonstrating how they transfer the Ramsey property and finite Ramsey degrees under broad, optimal conditions.

Findings

Established transfer of Ramsey property via semi-retractions.

Identified conditions for finite Ramsey degrees transfer.

Compared semi-retractions with pre-adjunctions in category theory.

Abstract

We investigate the notion of a semi-retraction between two first order structures (in typically different signatures) that was introduced by the second author as a link between the Ramsey property and generalized indiscernible sequences. We further these connections between combinatorics and model theory, and look at semi-retractions through a new lens establishing transfers of the Ramsey property and finite Ramsey degrees under quite general conditions that are optimal as demonstrated by counterexamples. Finally, we compare semi-retractions to the category theoretic notion of a pre-adjunction.