The Horizontal Direction
Tristan van der Vlugt
TL;DR
The paper surveys cardinal characteristics of the higher Cichoń diagram on the higher Baire space ${}^\kappa\kappa$ for regular $\kappa$ with $2^{<\kappa}=\kappa$, focusing on horizontal separations such as $\mathrm{add}(\mathcal M_\kappa)<\mathrm{non}(\mathcal M_\kappa)$ and $\mathrm{cov}(\mathcal M_\kappa)<\mathrm{cof}(\mathcal M_\kappa)$. It compares classical forcing techniques with their higher analogues, explaining why horizontal separation is substantially harder in the higher context and detailing how vertical separation generalizes more readily. The survey analyzes higher Cohen, Hechler, localisation, Sacks, random, Laver, Mathias, Miller, and eventual-difference forcings, highlighting where higher properties (such as the $H$-Sacks or $h$-Laver properties) succeed or fail and the implications for localisation and domination invariants. It also surveys partial results regarding higher random forcing, Borel-type conjectures at $\kappa$, and Milller-type forcings, and identifies key open questions, including preservation under iterations and potential side-by-side constructions, that must be resolved to realize horizontal separations in a broad higher setting.
Abstract
We give a survey of cardinal charcteristics of the higher Cichoń diagram defined on the higher Baire space ${}^κκ$ for $κ$ regular with $2^{<κ}=κ$. Specifically, we will compare consistency proofs from the classical Cichoń diagram with various well-known forcing notions to similar constructions generalised to the higher Cichoń diagram. We are especially interested in separation in a horizontal direction, that is, the consistency of $\mathrm{add}(\mathcal M_κ)<\mathrm{non}(\mathcal M_κ)$ and of $\mathrm{cov}(\mathcal M_κ)<\mathrm{cof}(\mathcal M_κ)$. We will have a look at (higher analogues of) Cohen, Hechler, localisation, eventually different, Sacks, random, Laver, Mathias and Miller forcing, and their effect on the cardinal characteristics of the higher Cichoń diagram.