Edge distribution and density in the characteristic sequence

The characteristic sequence of hypergraphs 〈 P n : n < ω 〉 associated to a formula φ ( x ; y ) , introduced in Malliaris (2010) [5], is defined by P n ( y 1 , … , y n ) = ( ∃ x ) ⋀ i ≤ n φ ( x ; y i ) . 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 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 S O P 3 ; this sheds light on the interplay of independence and order in unstable theories.