Edge distribution and density in the characteristic sequence

The characteristic sequence of hypergraphs 〈Pn : n < ω〉 associated to a formula φ(x; y), introduced in [5], is defined by Pn(y1, . . . yn) = (∃x) ∧ i≤n φ(x; yi). We continue 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 Pn (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 SOP3; this sheds light on the interplay of independence and order in unstable theories.