Saturated filters at successors of singulars, weak reflection and yet another weak club principle

More Jonsson algebras

We prove that on many inaccessible there is a Jonsson algebra, so e.g. the first regular Jonsson cardinal lambda which is lambda x omega-Mahlo. We give further restrictions on successor of singulars which are Jonsson cardinals. E.g. there is a Jonsson algebra on the successor of beth_omega. Lastly, we give further information on guessing of clubs.

More rigid ideals

We extend prior results of Cody-Eskew, showing the consistency of GCH with the statement that for all regular cardinals $κ\leq λ$, where $κ$ is the successor of a regular cardinal, there is a rigid saturated ideal on $\mathcal{P}_κλ$. We also show the consistency of some instances of rigid saturated ideals on $\mathcal{P}_κλ$ where $κ$ is the successor of a singular cardinal.

后继基数上的超滤与Tukey序

Ultrafilters over Successor Cardinals and the Tukey Order

该研究聚焦正则不可数基数（尤其$\omega_1$）上的超滤与Tukey序，证明了相关断言的ZFC独立性，给出新型超滤构造，分析了特定公理下超滤的序性质并得到多个相关结果。

Jensen's diamond principle and its relatives

We survey some recent results on the validity of Jensen's diamond principle at successor cardinals. We also discuss weakening of this principle such as club guessing, and anti-diamond principles such as uniformization. A collection of open problems is included.

Compactness versus hugeness at successor cardinals

If $κ$ is regular and $2^{<κ}\leqκ^+$, then the existence of a weakly presaturated ideal on $κ^+$ implies $\square^*_κ$. This partially answers a question of Foreman and Magidor about the approachability ideal on $ω_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $ω_2$ such that $\mathcal{P}(ω_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.

A note on strong negative partition relations

We analyze a natural function definable from a scale at a singular cardinal, and using this function we are able to obtain quite strong negative square-brackets partition relations at successors of singular cardinals. The proof of our main result makes use of club-guessing, and as a corollary we obtain a fairly easy proof of a difficult result of Shelah connecting weak saturation of a certain club-guessing ideal with strong failures of square-brackets partition relations. We then investigate the strength of weak saturation of such ideals and obtain some results on stationary reflection.

$P_κ(λ)$的$μ$-俱乐部：天堂中的乐园

$μ$-Clubs of $P_κ(λ)$ : Paradise in heaven

该研究在无大基数内模型等条件下，证明了$P_κ(λ)$与$P_κ(u(κ,λ))$上的$μ$-俱乐部滤子同构，且其对偶理想不满足特定饱和性，推进了无穷基数组合相关研究。

Prikry-type forcings after collapsing a huge cardinal

Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer principles for chromatic numbers of graphs. In this paper, we study these in the case of the successors of singular cardinals. In particular, we show that Prikry forcing preserves the centeredness of ideals but kills the layeredness. We also study $\binom{μ^{++}}{μ^{+}}\to\binomκ{μ^{+}}_μ$ and $\mathrm{Tr}_{\mathrm{Chr}}(μ^{+++},μ^{+})$ in the extension by Prikr…

Destroying Saturation while Preserving Presaturation at an Inaccessible; an Iterated Forcing Argument

We prove that a large class of presaturated ideals at inaccessible cardinals can be de-saturated while preserving their presaturation, answering both a question of Foreman and of Cox and Eskew. We do so by iterating a generalized version of Baumgartner and Taylor's forcing to add a club with finite conditions along an inaccessible cardinal, and invoking Foreman's Duality Theorem.