TREE FORCING AND DEFINABLE MAXIMAL INDEPENDENT SETS IN HYPERGRAPHS
TREE FORCING AND DEFINABLE MAXIMAL INDEPENDENT SETS IN HYPERGRAPHS
复制标题
超图中的树强制和可定义最大独立集
DOI：
10.1017/jsl.2022.36
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
J. Schilhan
中科院分区：
文献类型：
--
作者：
J. Schilhan
本文证明了在L上用可数支集迭代或Sacks有限积强迫或分裂强迫后，Polish空间上的每个解析超图都允许一个
$\mathbf {\Delta }^1_2$
极大独立集这推广了Schrittesser的一个较早的结果（见[25]）。作为主要应用程序，我们获得了以下内容的一致性
$\mathfrak {r} = \mathfrak {u} = \mathfrak {i} = \omega _2$
再加上一个
$\Delta ^1_2$
超滤器
$\Pi ^1_1$
极大独立族，和
$\Delta ^1_2$
Hamel基这解决了Brendle，Fischer和Khomskii [5]和作者[23]的公开问题。我们还在ZFC中显示，
$\mathfrak {d} \leq \mathfrak {i}_{cl}$
”[5]这是另一个问题。
Abstract 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
$\mathbf {\Delta }^1_2$
maximal independent set. This extends an earlier result by Schrittesser (see [25]). As a main application we get the consistency of
$\mathfrak {r} = \mathfrak {u} = \mathfrak {i} = \omega _2$
together with the existence of a
$\Delta ^1_2$
ultrafilter, a
$\Pi ^1_1$
maximal independent family, and a
$\Delta ^1_2$
Hamel basis. This solves open problems of Brendle, Fischer, and Khomskii [5] and the author [23]. We also show in ZFC that
$\mathfrak {d} \leq \mathfrak {i}_{cl}$
, addressing another question from [5].