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INFINITE GAMES AND RAMSEY PROPERTIES OF 
$F_\sigma $ IDEALS
Part of:
Set theory
Published online by Cambridge University Press: 17 February 2025
Abstract
In this work, we investigate various combinatorial properties of Borel ideals on countable sets. We extend a theorem presented in [13] and identify an 
$F_\sigma $ tall ideal in which player II has a winning strategy in the Cut and Choose Game, thereby addressing a question posed by J. Zapletal. Additionally, we explore the Ramsey properties of ideals, demonstrating that the random graph ideal is critical for the Ramsey property when considering more than two colors. The previously known result for two colors is extended to any finite number of colors. Furthermore, we comment on the Solecki ideal and identify an 
$F_\sigma $ tall K-uniform ideal that is not equivalent to 
$\mathcal {ED}_{\text {fin}}$, thereby addressing a question from M. Hrušák’s work [10].
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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