Tukey order among $F_{\sigma}$ ideals

TL;DR

This paper explores the Tukey order among $F_\sigma$ ideals, introducing flat and gradually flat ideals, and establishing a dichotomy theorem that characterizes their structural properties and relationships.

Contribution

It introduces the concepts of flat and gradually flat ideals, providing a dichotomy theorem and characterizations of gradual flatness within the Tukey order framework.

Findings

No nontrivial $F_\sigma$ ideal is Tukey below a $G_\delta$ ideal of compact sets.

Gradually flat ideals are exactly those flat ideals Tukey below the ideal of density zero sets.

Diverse characterizations of gradual flatness using Tukey reductions and game-theoretic approaches.

Abstract

We investigate the Tukey order in the class of $F_\sigma$ ideals of subsets of $\omega$. We show that no nontrivial $F_\sigma$ ideal is Tukey below a $G_\delta$ ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For example, we show that gradually flat ideals are precisely those flat ideals that are Tukey below the ideal of density zero sets.