Small infinite partitions and other features of the Nowhere Centered ideal

TL;DR

This paper introduces the nowhere dense ideal, explores its unique combinatorial properties, and compares it with other definable quotients, highlighting its large partitions and structural features.

Contribution

It defines the nowhere dense ideal and analyzes its combinatorial and partition properties, revealing contrasts with other known definable quotients.

Findings

The quotient has countable partitions.

It can have partitions of size .

The ideal exhibits unique combinatorial features.

Abstract

The \textit{nowhere dense ideal} $\mathcal{NC}$ is introduced. It is a coanalytic ideal of $\omega\times\omega$ whose defining characteristic is that the sets of the form $X\times Y$, where $X,Y$ are infinite subsets of $\omega$, are dense in the quotient $\mathcal{P}(\omega\times\omega)\slash\mathcal{NC}$. This quotient has countable partitions and consistently has partitions of size $\omega_1$ while $\mathfrak{a}>\omega_1$. This represents a huge contrast with other definable quotients. Other combinatorial features of this ideal are presented, as well as some results on a family of similar, higher dimensional ideals.