Superclub, splitting, separating statements

TL;DR

This paper proves that the superclub principle implies certain cardinal equalities and uses this to distinguish it from tiltan at successors of large cardinals, employing Galvin's property for separation results.

Contribution

It establishes the implication of superclub on cardinal characteristics and separates superclub from tiltan at successors of large cardinals using advanced combinatorial techniques.

Findings

Superclub implies =1 at certain cardinals.

Superclub and tiltan are separated at successors of large cardinals.

Galvin's property is used to distinguish between combinatorial principles.

Abstract

We prove that superclub implies $\mathfrak{s}=\aleph_1$. More generally, superclub at a successor of a weakly compact cardinal implies $\mathfrak{s}_\kappa=\kappa^+$. Based on this statement, we separate tiltan from superclub at a successor of a supercompact cardinal. We use Galvin's property in order to separate tiltan from superclub at successors of both regular and singular cardinals.