论文标题
在超图中的树强迫和可定义的最大独立集
Tree forcing and definable maximal independent sets in hypergraphs
论文作者
论文摘要
我们表明，在强迫可数的支撑迭代或麻袋的有限产物或拆分迫使$ l $以上的产品之后，波兰空间上的每个分析超图都承认$ \mathbfΔ^1_2 $最大独立套件。作为一个主要应用程序，我们获得$ \ Mathfrak {r} = \ Mathfrak {u} = \ Mathfrak {i} =ω_2$的一致性，以及存在$δ^1_2 $ ultrafilter，$Δ^1_2 $ ultrafilter，$π^1_1 $最大独立家庭和$δ^1_2 $Δ^1_2 $ HAMEL基础。这解决了Brendle，Fischer和Khomskii和作者的开放问题。我们还在ZFC中显示$ \ mathfrak {d} \ leq \ mathfrak {i} _ {cl} $。
We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over $L$, every analytic hypergraph on a Polish space admits a $\mathbfΔ^1_2$ maximal independent set. As a main application we get the consistency of $\mathfrak{r} = \mathfrak{u} = \mathfrak{i} = ω_2$ together with the existence of a $Δ^1_2$ ultrafilter, a $Π^1_1$ maximal independent family and a $Δ^1_2$ Hamel basis. This solves open problems of Brendle, Fischer and Khomskii and the author. We also show in ZFC that $\mathfrak{d} \leq \mathfrak{i}_{cl}$.
