用独立的小值域族饱和随机图
Saturating the random graph with an independent family of small range
摘要：
受Keisler的序的启发，我们证明了对于$I$上的一类正则滤子$D$，$|I| = λ> \aleph_0$，$P(I)/\de$几乎没有自由度的事实（如通过任何最大反链的大小为$<λ$的事实所测量，或者甚至是可数的）并不阻止将$D$扩展到使随机图的超幂饱和的$I$上的超滤波器$D_1$。“饱和”意味着当M是随机图理论的模型时，$M^I/\de_1$是$λ^+$ -饱和的。这对于稳定的理论是正确的，而对于非简单和非低的理论是错误的。这一结果和证明中引入的技术催化了作者随后关于简单不稳定理论的Keisler阶的工作。介绍，其中包括一部分写的模型理论家和一部分写的集理论家，讨论我们目前的计划和相关的结果。
英文摘要：
Motivated by Keisler's order, a far-reaching program of understanding basic model-theoretic structure through the lens of regular ultrapowers, we prove that for a class of regular filters $D$ on $I$, $|I| = λ> \aleph_0$, the fact that $P(I)/\de$ has little freedom (as measured by the fact that any maximal antichain is of size $<λ$, or even countable) does not prevent extending $D$ to an ultrafilter $D_1$ on $I$ which saturates ultrapowers of the random graph. "Saturates" means that $M^I/\de_1$ is $λ^+$-saturated whenever M is a model of the theory of the random graph. This was known to be true for stable theories, and false for non-simple and non-low theories. This result and the techniques introduced in the proof have catalyzed the authors' subsequent work on Keisler's order for simple unstable theories. The introduction, which includes a part written for model theorists and a part written for set theorists, discusses our current program and related results.
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