## Journal of Symbolic Logic

### The characteristic sequence of a first-order formula

M. E. Malliaris

#### Abstract

For a first-order formula φ(x;y) we introduce and study the characteristic sequence 〈 Pn: n < ω 〉 of hypergraphs defined by Pn(y1,…,yn) := (∃ x) ⋁i ≤ n φ(x;yi). We show that combinatorial and classification theoretic properties of the characteristic sequence reflect classification theoretic properties of φ and vice versa. The main results are a characterization of NIP and of simplicity in terms of persistence of configurations in the characteristic sequence. Specifically, we show that some tree properties are detected by the presence of certain combinatorial configurations in the characteristic sequence while other properties such as instability and the independence property manifest themselves in the persistence of complicated configurations under localization.

#### Article information

Source
J. Symbolic Logic, Volume 75, Issue 4 (2010), 1415-1440.

Dates
First available in Project Euclid: 4 October 2010

https://projecteuclid.org/euclid.jsl/1286198155

Digital Object Identifier
doi:10.2178/jsl/1286198155

Mathematical Reviews number (MathSciNet)
MR2767977

Zentralblatt MATH identifier
1220.03019

#### Citation

Malliaris, M. E. The characteristic sequence of a first-order formula. J. Symbolic Logic 75 (2010), no. 4, 1415--1440. doi:10.2178/jsl/1286198155. https://projecteuclid.org/euclid.jsl/1286198155