Adding the constant evasion and constant prediction numbers to Cichoń's maximum

Abstract

Let ${\mathfrak{e}}_2^{\mathsf{const}}$ be the constant evasion number, that is, the size of the least family $F\subseteq {}^{\ddot{I}}‰2$ of reals such that for each predictor $\ddot{I}€:{}^{<\ddot{I}‰}2\to 2$ there is $x\in F$ which is not constantly predicted by $\ddot{I}€$; and let ${\mathfrak{v}}_2^{\mathsf{const}}$ be the constant prediction number, that is, the size of the least family ${\hat{I}}_2$ of functions $\ddot{I}€:{}^{<\ddot{I}‰}2\to 2$ such that for each $x\in {}^{\ddot{I}}‰2$ there is $\ddot{I}€\in {\hat{I}}_2$ 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.

32 pages, 6 figures