The Horizontal Direction

Abstract

We give a survey of cardinal charcteristics of the higher Cichoń diagram defined on the higher Baire space ${}^{\hat{I}}º\hat{I}º$ for $\hat{I}º$ regular with $2^{<\hat{I}º}=\hat{I}º$. 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 $\mathrm{add}({\mathcal{M}}_{\hat{I}}º)<\mathrm{non}({\mathcal{M}}_{\hat{I}}º)$ and of $\mathrm{cov}({\mathcal{M}}_{\hat{I}}º)<\mathrm{cof}({\mathcal{M}}_{\hat{I}}º)$. 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.

30 pages, survey article, submitted to RIMS kokyuroku