Infinite Games and Ramsey Properties of $F_σ$ Ideals
José de Jesús Pelayo Gómez
TL;DR
This work extends Ramsey-type analysis for definable ideals on $\omega$, proving that tall $F_\sigma$ ideals can exhibit surprising multi-color Ramsey thresholds and strategic behavior in Cut and Choose games. By constructing explicit ideals like $\mathcal{PC}$ and $\mathcal{ED}_m$ and introducing color-augmented random-graph frameworks ($R_k^n$, $\mathcal{R}_{k,l}^n$), the authors connect game-theoretic winning strategies to Katětov-order relations and provide a nuanced map of Ramsey properties across colors. The study also broadens the landscape with the Solecki-based $\mathcal{S}_\omega$ and a new $K$-uniform ideal $\mathcal{K}$, clarifying how these ideals fit into the broader hierarchy and addressing questions posed by Zapletal and Hrušák. Overall, the results deepen our understanding of the combinatorial structure of definable ideals and their Ramsey-theoretic boundaries, with implications for tallness, uniformity, and ordering in the Katětov framework.
Abstract
In this work, we investigate various combinatorial properties of Borel ideals on countable sets. We extend a theorem presented in M. Hrušák, D. Meza-Alcántara, E. Thümmel, and C. Uzcátegui, \emph{Ramsey Type Properties of Ideals}, and identify an $F_σ$ 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 $F_σ$ tall $K$-uniform ideal that is not equivalent to $\mathcal{ED}_{\text{fin}}$, thereby addressing a question from Michael Hrušák.