马丁公理与$ω_1^2 \to (ω_1^2, 3)^2$
Martin's axiom and $ω_1^2 \longrightarrow (ω_1^2, 3)^2$
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中文总结 AI 辅助
该研究从CH和Hajnal着色出发，通过σ-中心力迫的有限支持迭代构造出满足$MA_{ω_1}$（σ-中心）等条件且$ω_1^2\nrightarrow(ω_1^2,3)^2$成立的模型，同时提出并证明了相关力迫保持原理。
中文摘要 AI 辅助
从连续统假设（CH）和Hajnal着色出发，我们证明σ-中心力迫的标准有限支持迭代可给出$MA_{ω_1}$（σ-中心）+$2^{ℵ_0}=ℵ_2$+$ω_1^2\nrightarrow(ω_1^2,3)^2$的模型。我们还分离出一条简单的力迫保持原理：在任何大小至多为$ω_1$的子族均可表为连通集的可数并的偏序集力迫后，原模型的同一着色仍为见证。在$MA_{ω_1}$下，每个c.c.c.力迫都具有该局部性质，因此该模型中所有现有见证都被每个c.c.c.力迫保持。
英文摘要
Starting with CH and Hajnal's coloring, we show that a standard finite-support iteration of $σ$-centered forcing notions gives a model of $ MA_{ω_1}(σ$-centered$)+2^{\aleph_0}=\aleph_2 +ω_1^2\nrightarrow(ω_1^2,3)^2.$ We also isolate a simple forcing-preservation principle: the same ground-model coloring remains a witness after forcing with any poset whose subfamilies of size at most $ω_1$ are countable unions of linked sets. Under $\text{MA}_{ω_1}$, every c.c.c. forcing has this local property, so every existing witness is preserved by every c.c.c. forcing over that model.
