中科院数学与系统科学研究院

数学研究所

数理逻辑讨论班

报告人：Dr. Victor Torres Perez（TU Wien)

题  目：Diamonds, games and cardinal invariants

时  间：2018.10.11（星期四）, 14:00--16:00

地  点：数学院南楼N820室

摘  要: On one hand, we prove that $\mathrm {WRP}$ and saturation of the ideal $\mathrm {NS}_{\omega_1}$ together imply $\Diamond\{a\in [\lambda]^{\omega_1}: \mathrm{cof}\left( \sup(a)\right)=\omega_1 \}$, for all regular $\lambda\geq \aleph_2$. On the other hand, in a joint work with Brendle and Hrusak, we consider a weak parametrized versions of the diamond principle which imply game versions of cardinal invariants $\mathfrak t$, $\mathfrak u$ and $\mathfrak a$. We show that the standard proof that parametrized diamond principles prove that the cardinal invariants are small actually show that their game counterparts are small. We show that $\mathfrak t<\mathfrak t_{game}$ and  $\mathfrak u<\mathfrak u_{game}$ are both relatively consistent with the ZFC. The corresponding question for $\mathfrak a$ remains open.