The Gamified Kat\v{e}tov order is not linear (in fact, very much not so)

TL;DR

This paper explores the properties of the Gamified Kat97ov order, showing it is complex and contains a large antichain, with connections to Ramsey theory and implications for computability hierarchies.

Contribution

It demonstrates that the Gamified Kat97ov order embeds 9(9)/Fin, contains a continuum-sized antichain, and links to Ramsey theory and Weihrauch degrees.

Findings

The Gamified Kat97ov order is not linear and contains a large antichain.

It embeds the Boolean algebra 9(9)/Fin, showing high complexity.

Connections are established between the order, Ramsey theory, and computability hierarchies.

Abstract

Recently, the authors introduced the Gamified Kat\v{e}tov order on filters over $\omega$. This was shown to be strictly coarser than the classical Kat\v{e}tov order, and in fact collapses all MAD families to a single equivalence class. In the opposite direction, the present paper shows that the Gamified Kat\v{e}tov order also embeds $\mathcal{P}(\omega)/\mathrm{Fin}$, and thus contains an antichain of size continuum. The analysis brings into focus some interesting connections with Ramsey theory. As part of a broader programme investigating the interplay between combinatorial and computable complexity, we then apply our construction to produce a large new family of non-modest degrees in the extended Weihrauch hierarchy, which arise from associated effective subtoposes.