理想化Miller空集的基数不变量

Splitting families of sets in ZFC

Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,ρ)$ in which $n$ is finite and $ρ$ is infinite. An extension of Miller's theorem is proved here in ZFC for pairs of cardinals $(ν,ρ)$ in which $ν$ is arbitrary and $ρ\ge \beth_\om(ν)$. The proof uses a new general method that is based on Shelah's revises Generalized Continuum Hypothesis theorem. Upper bounds on conflict-free coloring numbers of families of sets and a general comparison theorem follow as corollaries of the main theorem. Other corollaries eliminate the use of additional axioms from splitting theorems due to E…

Forcing Diamond 及其在可迭代性中的应用

Forcing Diamond and Applications to Iterability

本文证明高阶Sacks力迫与club Miller力迫可添加diamond序列，并肯定回答了κ-Sacks力迫的κ-支撑迭代不坍缩κ+且为κ-proper的长期问题，结果适用于其他高阶树力迫。

Forcing, games and families of closed sets

We propose a new, game-theoretic, approach to the idealized forcing, in terms of fusion games. This generalizes the classical approach to the Sacks and the Miller forcing. For definable ($\mathbfΠ^1_1$ on $\mathbfΣ^1_1) $σ$-ideals we show that if a $σ$-ideal is generated by closed sets, then it is generated by closed sets in all forcing extensions. We also prove an infinite-dimensional version of the Solecki dichotomy for analytic sets. Among examples, we investigate the $σ$-ideal $\E$ generated by closed null sets and $σ$-ideals connected with not piecewise continuous functions.

贫集与闭测度零集的平移组合学

Combinatorics of translations of meager and closed measure zero sets

研究贫集与闭测度零集平移的组合性质，探讨相关Borel猜想，获得贫理想可加性新刻画，并证明若干基数不变量关系。

Combinatorial properties of classical forcing notions

We discuss the effect of adding a single real (for various forcing notions adding reals) on cardinal invariants associated with the continuum (like the unbounding or the dominating number or the cardinals related to measure and category on the real line). For random and Cohen forcing, this question was investigated by Cicho'n and Pawlikowski; for Hechler forcing, by Judah, Shelah and myself. We show here: (1) adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size omega_1; (2) Laver and Mathias forcing collapse the dominating number …

Special subsets of the generalized Cantor space $2^κ$ and generalized Baire space $κ^κ$

In this paper, we are interested in parallels to the classical notions of special subsets in $\R$ defined in the generalized Cantor and Baire spaces ($2^κ$ and $κ^κ$). We consider generalizations of the well-known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, $σ$-sets, $γ$-sets, sets with Menger, Rothberger or Hurewicz property, but also of some less-know classes like $X$-small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller and Laver-null sets. We notice that many classical theorems regarding these classes c…

有秩强制与广义博雷尔层次结构的长度

Ranked Forcing and the Length of Generalized Borel Hierarchies

本文研究了广义博雷尔层次结构的长度，通过扩展α-强制框架并构造特定模型，揭示了多个子空间上层次结构的非平凡星座，并推导了well-founded树的精确复杂度。