Adding the constant evasion and constant prediction numbers to Cichoń's maximum
Miguel A. Cardona, Miroslav Repický, Saharon Shelah
TL;DR
This work shows that the constant evasion number $\mathfrak{e}_2^{\mathrm{const}}$ and the constant prediction number $\mathfrak{v}_2^{\mathrm{const}}$ can be added to Cichoń's maximum with distinct values. It develops a robust forcing framework based on relational systems, Tukey connections, UF-limits, and simple matrix iterations, together with the new $\Pr^2_{\bar{n}^*}(\lambda)$ property to keep $\mathrm{add}(\mathcal{N})$ small in extensions. Through intricate FS and matrix-iteration constructions, the authors realize constellations that separate the right-hand side of Cichoń's diagram and then extend the maximum to incorporate $\mathfrak{e}^{\mathrm{const}}_2$ and $\mathfrak{v}^{\mathrm{const}}_2$, using elementary submodel arguments to control the forcing impact. The results provide both left- and right-hand side separations with these constants and raise several open questions about further refinements and consistency, highlighting the versatility of UF-limits and goodness notions in complex cardinal-characteristic forcing.
Abstract
Let $\mathfrak{e}^\mathsf{const}_2$ be the constant evasion number, that is, the size of the least family $F\subseteq{}^ω2$ of reals such that for each predictor $π\colon {}^{<ω}2\to 2$ there is $x\in F$ which is not constantly predicted by $π$; and let $\mathfrak{v}_2^\mathsf{const}$ be the constant prediction number, that is, the size of the least family $Π_2$ of functions $π\colon {}^{<ω}2\to 2$ such that for each $x\in{}^ω2$ there is $π\inΠ_2$ that predicts constantly $x$. In this work, we show that the constant evasion number $\mathfrak{e}_2^{\mathrm{cons}}$ and the constant prediction number $\mathfrak{v}_2^\mathsf{const}$ can be added to Cichoń's maximum with distinct values.