Halfway New Cardinal Characteristics

TL;DR

This paper introduces nine new cardinal characteristics based on asymptotic density, exploring their bounds and consistency results related to classical characteristics like , , and , and investigates their possible values.

Contribution

It defines new cardinal characteristics using asymptotic density and establishes bounds and consistency results, expanding the understanding of intersection properties of infinite sets.

Findings

Proves the consistency of  < _{1/2}

Provides bounds for the new characteristics

Analyzes possible values of _{1/2}

Abstract

Based on the well-known cardinal characteristics $\mathfrak{s}$, $\mathfrak{r}$ and $\mathfrak{i}$, we introduce nine related cardinal characteristics by using the notion of asymptotic density to characterise different intersection properties of infinite sets. We prove several bounds and consistency results, e. g. the consistency of $\mathfrak{s} < \mathfrak{s}_{1/2}$, as well as several results about possible values of $\mathfrak{i}_{1/2}$.