The Horizontal Direction

Abstract

We give a survey of cardinal charcteristics of the higher Cicho\'n diagram defined on the higher Baire space ${}^\kappa\kappa$ for $\kappa$ regular with $2^{<\kappa}=\kappa$. Specifically, we will compare consistency proofs from the classical Cicho\'n diagram with various well-known forcing notions to similar constructions generalised to the higher Cicho\'n diagram. We are especially interested in separation in a horizontal direction, that is, the consistency of $\mathrm{add}(\mathcal M_\kappa)<\mathrm{non}(\mathcal M_\kappa)$ and of $\mathrm{cov}(\mathcal M_\kappa)<\mathrm{cof}(\mathcal M_\kappa)$. 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 Cicho\'n diagram.