Infinite Games and Ramsey Properties of $F_{\sigma }$ Ideals

Abstract

We study combinatorial properties of Borel ideals on countable sets. We construct a tall $F_{\sigma }$ ideal for which player II has a winning strategy in the cut-and-choose game, answering a question of J. Zapletal. We also generalize the characterization of Ramsey properties by the random graph ideal to ideals associated with random colorings of finite subsets. Extending a construction of Hrušák, Meza-Alcántara, Thümmel, and Uzcátegui, we obtain, for each $m\in \omega$, a tall $F_{\sigma }$ ideal ${\mathcal{E}\mathcal{D}}_m$ satisfying $\omega \to ({\mathcal{E}\mathcal{D}}_m^{+}{)}_{m+1}^2$ but not $\omega \to ({\mathcal{E}\mathcal{D}}_m^{+}{)}_{m+2}^2$. We also construct a tall $F_{\sigma \delta }$ ideal ${\mathcal{S}}_{\omega }$ extending the Solecki ideal, with ${\mathcal{S}}_{\omega }^{+}\to (\omega ,{\mathcal{S}}_{\omega }^{+}{)}_2^2$ but ${\mathcal{S}}_{\omega }^{+}\nrightarrow ({\mathcal{S}}_{\omega }^{+}{)}_2^2$, and derive a Ramsey-type property of the Solecki ideal. Finally, we construct a tall $F_{\sigma }$ $K$-uniform ideal that is not Katětov-equivalent to ${\mathcal{E}\mathcal{D}}_{\mathrm{fin}}$, answering a question of M. Hrušák.

arXiv comment31 pages. Revised proofs and exposition; added a negative answer to Question 2.1. Numbering of results preserved