Evasion and prediction. V: Unsymmetric game ideals, constant prediction, and strong porosity ideals (Q2425337)

For a~natural number \(k\ge2\) let \(\mathcal{D}_k\) denote the family of all sets \(A\subseteq k^\omega\) for which there is \(g:k^{<\omega}\to k\) such that for all \(x\in A\), \(x(n)\ne g(x{\upharpoonright}n)\) holds for all but finitely many \(n\in\omega\). The sets in \(\mathcal{D}_k\) are sets with winning strategies in unsymmetric games on \(k^\omega\). The authors prove \(\mathrm{cov}(\mathcal{D}_k)\ge\mathrm{non}(\mathcal{M})\) and \(\mathrm{non}(\mathcal{D}_k)\le\mathrm{cov}(\mathcal{M})\) where \(\mathcal{M}\) and \(\mathcal{N}\) denote the meager and null ideals on the Cantor space, respectively. Consequently, \(\mathrm{non}(\mathcal{D}_k)\le\mathrm{cov}(\mathcal{D}_k)\) for all~\(k\) which answers a~question of \textit{L. Newelski} and \textit{A. Roslanowski} [Proc. Am. Math. Soc. 117, No. 3, 823--831 (1993; Zbl 0778.03016)]. A~predictor is a~function \(\pi:2^{<\omega}\to2\). A~predictor~\(\pi\) \(k\)-constantly predicts a~real \(x\in2^\omega\), if for all but finitely many intervals \(I\subseteq\omega\) of length~\(k\) there is \(i\in I\) such that \(x(i)=\pi(x{\upharpoonright}i)\). The \(k\)-constant prediction number \(\mathfrak{v}_2^\mathrm{const}(k)\) is the least size of a~set~\(\Pi\) of predictors such that every \(x\in2^\omega\) is \(k\)-constantly predicted by some \(\pi\in\Pi\). Dually, the \(k\)-constant evasion number \(\mathfrak{e}_2^\mathrm{const}(k)\) is the least size of a set of reals \(X\subseteq2^\omega\) such that for every predictor~\(\pi\) there is \(x\in X\) that is not \(k\)-constantly predicted by~\(\pi\). The authors describe the connection between strong porosity in \(2^\omega\) and the constant prediction. They prove that the uniformity and the covering numbers of strong porosity agree with the constant evassion and prediction numbers, respectively. They clarify the relation between the \(k\)-porosity and the ideals~\(\mathcal{D}_{2^k}\). They answer a question of \textit{S. Kamo} [J. Math. Soc. Japan 53, No. 1, 35--57 (2001; Zbl 0971.03043)] by proving that \(\mathfrak{v}_2^\mathrm{const}\ge\mathrm{non}(\mathcal{M})\). For Part IV see [the author and \textit{S. Shelah}, Arch. Math. Logic 42, No. 4, 349--360 (2003; Zbl 1029.03036)].