平稳公共邻域与划分假设
Stationary common-neighborhood properties and partition hypotheses
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中文总结 AI 辅助
本文从大基数出发力迫得到平稳公共邻域性质并改进划分假设的上界，同时证明划分假设在Tukey序下具有传递性。
中文摘要 AI 辅助
从一个$\omega_1$-Ramsey基数出发，我们力迫使得$[\omega_2]^2$的每个可数染色都有一个平稳集$X$和一种颜色$i$，使得$X$的每个有限子集在$X$中都有平稳多个颜色为$i$的公共邻域。由此得到的单色图在任意非平稳删除后直径至多为2，回答了Hrušák--Shelah--Zhang的一个问题。从一个Ramsey基数出发，我们力迫划分假设$\operatorname{PH}_1(\omega_2)$，改进了Bannister--Bergfalk--Moore--Todorcevic记录的大基数上界。证明共享一个局部正规理想构造。平稳论证使用可数封闭模型，而Ramsey论证使用逐点融合和超滤子的外部可数完备性。最后，对于非空有向拟序$P\leq_TQ$，每个$n<\omega$，以及每个基数$\lambda$，我们证明$\operatorname{PH}_n(Q,\lambda)$蕴含$\operatorname{PH}_n(P,\lambda)$，回答了同一作者提出的Tukey传递问题。
英文摘要
We use stationary common-neighborhood properties to study highly connected Ramsey relations and partition hypotheses. For weakly compact $κ$, $\operatorname{Coll}(ω_1,{<}κ)$ forces $ω_2\to_{\mathrm{hc},<5}(ω_2)^2_ω$ and $\operatorname{PH}_1(ω_2)$. If $κ$ is $T^{κ^+}_{ω_1}$-Ramsey, the same collapse forces that every countable coloring of $[ω_2]^2$ has a stationary set $X\subseteqω_2$ and a color $i$ such that every finite subset of $X$ has stationarily many color-$i$ common neighbors in $X$. From one weakly compact cardinal, we obtain a model of the ${<}5$-edge relation at $ω_3$ and $\operatorname{PH}_1(ω_3)$, in which $\check H^2(ω_3,A_d)\ne0$ for every nontrivial abelian group $A$. This separates $\operatorname{PH}_1(ω_3)$ from $\operatorname{PH}_2(ω_3)$, with the exact consistency strength of one weakly compact cardinal. We also show that $\operatorname{PH}_1(ω_2\timesω_5)$ is equiconsistent with two weakly compact cardinals.
发表机构
- University of California, Los Angeles（加州大学洛杉矶分校）
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