特征序列中的边缘分布和密度
Edge distribution and density in the characteristic sequence
摘要：
与[arXiv：0908.4111]中引入的公式$ϕ(x;y)$相关联的超图$<P_n : n<ω>$的特征序列由$P_n(y_1,... y_n) = (\exists x) \bigwedge_{i\leq n} ϕ(x;y_i)$定义。本文继续研究的特征序列，表明图论的技术，特别是Szemerédi的著名的正则引理，可以自然地应用到模型理论的复杂性通过特征序列的研究。具体而言，我们涉及分类理论性质的$ϕ$和$P_n$（被认为是公式）的组件之间的密度在Szemerédi经常分解图的特征序列。此外，我们使用Szemerédi正则性校准模型理论的独立性的概念，通过描述的独立性的星座集的深度，并表明，某些故障的深度意味着谢拉的强顺序属性$SOP_3$，这揭示了不稳定的理论中的独立性和顺序的相互作用。
英文摘要：
The characteristic sequence of hypergraphs $<P_n : n<ω>$ associated to a formula $ϕ(x;y)$, introduced in [arXiv:0908.4111], is defined by $P_n(y_1,... y_n) = (\exists x) \bigwedge_{i\leq n} ϕ(x;y_i)$. This paper continues the study of characteristic sequences, showing that graph-theoretic techniques, notably Szemerédi's celebrated regularity lemma, can be naturally applied to the study of model-theoretic complexity via the characteristic sequence. Specifically, we relate classification-theoretic properties of $ϕ$ and of the $P_n$ (considered as formulas) to density between components in Szemerédi-regular decompositions of graphs in the characteristic sequence. In addition, we use Szemerédi regularity to calibrate model-theoretic notions of independence by describing the depth of independence of a constellation of sets and showing that certain failures of depth imply Shelah's strong order property $SOP_3$; this sheds light on the interplay of independence and order in unstable theories.
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