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In their two talks J.M. Kuiper and O. Spinas will report on their joint work.
Spinas will give a survey, while Kuiper will present some proofs.
\begin{center}
\Large
Mycielski Ideals and Uniform Sacks Trees
\end{center}
\begin{abstract}
The Mycielski ideal $\mathfrak{C}_{X}$ ($X$ any set with $\vert X\vert\geq 2$) consists of all sets $A\subseteq X^\omega$ such that Player II has a winning strategy in the game $\Gamma_X(A,b)$ for every $b\in[\omega]^\omega$, where this game is defined such that both players produce some $x\in X^\omega$, $x(n)$ being chosen by Player II iff $n\in b$, and Player I wins iff $x\in A$.
For the variant $\mathfrak{P}_{X}$ one asks that Player II always has a winning strategy that is independent of Player I's moves, hence $\mathfrak{P}_{X}\subseteq \mathfrak{C}_{X}$.
There exist close relations between Mycielski ideals and tree forcing ideals, especially the ideal $u^0$ associated with uniform Sacks forcing and the Silver ideal $v^0$, as well as their finite dimensional versions $u^0_k,v^0_k$ ($2\leq k < \omega$).
Our main results which build on and are motivated by \cite{brendle}, \cite{kamo1993}, \cite{mycielski}, \cite{repicky2004}, \cite{Roslanowski1990}, \cite{Roslanowski1994}, \cite{shelahSteprans}, \cite{Spinas2016}, \cite{spinas2022} are the following:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $\mathrm{cof}(\mathfrak{C}_{X})$ and $\mathrm{cof}(\mathfrak{P}_{X})$ may be singular and their cofinalities are $>2^{\aleph_0}$, for every $X$ with $\vert X\vert\geq 2$.
\item $\mathfrak{P}_{k}\subsetneq v^0_k$ and $\mathrm{Con}(\mathfrak{C}_{k}\subsetneq u^0_k)$ ($2\leq k<\omega$).
\item $\mathrm{add} (u^0_k)\leq \mathfrak{b}$ ($2\leq k < \omega$) and $\mathrm{Con}(\forall 2\leq k <\omega\; \mathfrak{d} < \mathrm{add}(\mathfrak{C}_{k}))$
\item $\mathfrak{P}_{k}\leq_T\mathfrak{P}_{k+1}$ and $\mathfrak{C}_{k}\leq_T\mathfrak{C}_{k+1}$ ($2\leq k < \omega$).
\item $\mathrm{Con}(\mathrm{cov}(\mathfrak{C}_{k+1}) < \mathrm{cov}(\mathfrak{C}_{k}))$ ($2\leq k < \omega$) (compare this with the result in \cite{shelahSteprans} that $\mathrm{cov}(\mathfrak{P}_{k}) = \mathrm{cov}(\mathfrak{P}_{k+1})$ in $\mathrm{ZFC}$)
\item $\mathrm{Con}(\mathrm{cov}(\mathcal{M}) < \mathrm{add}(u^0_k))$ ($2\leq k<\omega$)
\end{enumerate}
\end{abstract}
\begin{filecontents*}[overwrite]{kuiperspinasbib.bib}
@article {brendle,
AUTHOR = {Brendle, J\"{o}rg},
TITLE = {Strolling through paradise},
JOURNAL = {Fund. Math.},
FJOURNAL = {Fundamenta Mathematicae},
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YEAR = {1995},
NUMBER = {1},
PAGES = {1--25},
ISSN = {0016-2736},
MRCLASS = {03E05 (03E35)},
MRNUMBER = {1354935},
MRREVIEWER = {James Baumgartner},
DOI = {10.4064/fm-148-1-1-25},
URL = {https://doi.org/10.4064/fm-148-1-1-25},
}
@article{kuiperSpinas,
AUTHOR = {{K}uiper, Jelle Mathis and {S}pinas, Otmar},
TITLE = {Mycielski Ideals and Uniform Trees},
}
@article {kamo1993,
AUTHOR = {Kamo, Shizuo},
TITLE = {Some remarks about {M}ycielski ideals},
JOURNAL = {Colloq. Math.},
FJOURNAL = {Colloquium Mathematicum},
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ISSN = {0010-1354},
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DOI = {10.4064/cm-65-2-291-299},
URL = {https://doi.org/10.4064/cm-65-2-291-299},
}
@article {mycielski,
AUTHOR = {Mycielski, Jan},
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URL = {https://doi.org/10.4064/cm-20-1-71-76},
}
@article {repicky2004,
AUTHOR = {Repick\'{y}, Miroslav},
TITLE = {Mycielski ideal and the perfect set theorem},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {132},
YEAR = {2004},
NUMBER = {7},
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MRCLASS = {03E15 (03E17 91A44)},
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URL = {https://doi.org/10.1090/S0002-9939-04-07360-5},
}
@article {shelahSteprans,
AUTHOR = {{S}helah, Saharon and {S}tepr\={a}ns, Juris},
TITLE = {The covering numbers of {M}ycielski ideals are all equal},
JOURNAL = {J. Symbolic Logic},
FJOURNAL = {The Journal of Symbolic Logic},
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YEAR = {2001},
NUMBER = {2},
PAGES = {707--718},
ISSN = {0022-4812},
MRCLASS = {03E35 (03E05 03E10)},
MRNUMBER = {1833473},
MRREVIEWER = {Marek Balcerzak},
DOI = {10.2307/2695039},
URL = {https://doi.org/10.2307/2695039},
}
@Article{Roslanowski1990,
Author = {Andrzej {Ros{\l}anowski}},
Title = {{On game ideals}},
FJournal = {{Colloquium Mathematicum}},
Journal = {{Colloq. Math.}},
ISSN = {0010-1354},
Volume = {59},
Number = {2},
Pages = {159--168},
Year = {1990},
Publisher = {Polish Academy of Sciences (Polska Akademia Nauk - PAN), Institute of Mathematics (Instytut Matematyczny), Warsaw},
Language = {English},
DOI = {10.4064/cm-59-2-159-168},
MSC2010 = {03E15 03E40 91A05},
Zbl = {0724.04003}
}
@article {Roslanowski1994,
AUTHOR = {Ros{\l}anowski, Andrezej},
TITLE = {Mycielski ideals generated by uncountable systems},
JOURNAL = {Colloq. Math.},
FJOURNAL = {Colloquium Mathematicum},
VOLUME = {66},
YEAR = {1994},
NUMBER = {2},
PAGES = {187--200},
ISSN = {0010-1354},
MRCLASS = {04A15 (03E05 03E40 90D44)},
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URL = {https://doi.org/10.4064/cm-66-2-187-200},
}
@article {Spinas2016,
AUTHOR = {Spinas, Otmar},
TITLE = {Silver trees and {C}ohen reals},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {211},
YEAR = {2016},
NUMBER = {1},
PAGES = {473--480},
ISSN = {0021-2172},
MRCLASS = {03E05 (03E17)},
MRNUMBER = {3474971},
MRREVIEWER = {Jan Kraszewski},
DOI = {10.1007/s11856-015-1279-0},
URL = {https://doi.org/10.1007/s11856-015-1279-0},
}
@article {spinas2022,
AUTHOR = {Spinas, Otmar},
TITLE = {Tukey reductions of nowhere {R}amsey to {S}ilver null sets},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {150},
YEAR = {2022},
NUMBER = {6},
PAGES = {2715--2727},
ISSN = {0002-9939},
MRCLASS = {03E05 (03E17 03E35 06A07)},
MRNUMBER = {4399284},
DOI = {10.1090/proc/15869},
URL = {https://doi.org/10.1090/proc/15869},
}
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