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

Takayuki Kihara; Ming Ng

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arXiv · 2605.21473

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

Takayuki Kihara·Ming Ng

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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.

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Takayuki Kihara, Ming Ng. 2026-05-20. The Gamified Kat\v{e}tov order is not linear (in fact, very much not so). https://arxiv.org/abs/2605.21473

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