论文标题
疯狂家庭的组合特性
Combinatorial properties of MAD families
论文作者
论文摘要
我们研究$ \ textsf {mad} $ family的一些强大组合属性。理想的$ \ Mathcal {i} $是Shelah-Steprāns，如果对于每个集合$ x \ subseteq {\ left [ω\ right]}^{<ω} $，则$ \ nathcal {i} $的元素都在$ x $中的每个集合或包含在$ x $中的每个成员，或者包含在$ x $中的每个成员。我们证明，只有在理想$ \ textsf {fin} \ times \ textsf {fin} $上方的katětov{fin} $之上，就证明了borel理想是Shelah-Steprāns。我们证明Shelah-Steprāns$ \ textsf {mad} $家庭具有强大的坚不可摧的属性（特别是它们既是Cohen又是随机的无限制）。我们还考虑了$ \ textsf {mad} $ family的其他一些强大的组合属性。最后，证明拥有$ \ mathrm {non}（\ Mathcal {M}）= {\ aleph} _ {1} $，并且没有Shelah-Steprānssize $ {\ alleph} _ {1} $是一致的。
We study some strong combinatorial properties of $\textsf{MAD}$ families. An ideal $\mathcal{I}$ is Shelah-Steprāns if for every set $X\subseteq{\left[ ω\right]}^{<ω}$ there is an element of $\mathcal{I}$ that either intersects every set in $X$ or contains infinitely many members of it. We prove that a Borel ideal is Shelah-Steprāns if and only if it is Katětov above the ideal $\textsf{fin}\times\textsf{fin}$. We prove that Shelah-Steprāns $\textsf{MAD}$ families have strong indestructibility properties (in particular, they are both Cohen and random indestructible). We also consider some other strong combinatorial properties of $\textsf{MAD}$ families. Finally, it is proved that it is consistent to have $\mathrm{non}(\mathcal{M}) = {\aleph}_{1}$ and no Shelah-Steprāns families of size ${\aleph}_{1}$.
