On the existence of large antichains for definable quasi-orders

TL;DR

This paper extends the understanding of perfect antichains in definable quasi-orders, providing new characterizations at various levels of the projective hierarchy and beyond, advancing the theory of descriptive set theory.

Contribution

It generalizes existing theorems to co-analytic quasi-orders and explores their validity at higher definable cardinals, broadening the scope of antichain existence results.

Findings

Generalization of Dilworth-style characterization to co-analytic quasi-orders

Establishment of analogous theorems at higher definable cardinals

Exploration of extensions beyond the first projective level

Abstract

We generalize Harrington-Marker-Shelah's Dilworth-style characterization of the existence of non-empty perfect antichains to co-analytic quasi-orders, establish the analogous theorem at the next definable cardinal, and consider generalizations beyond the first level of the projective hierarchy.