## Notre Dame Journal of Formal Logic

### On $n$-Dependence

#### Abstract

In this article, we develop and clarify some of the basic combinatorial properties of the new notion of $n$-dependence (for $1\leq n\lt \omega$) recently introduced by Shelah. In the same way as dependence of a theory means its inability to encode a bipartite random graph with a definable edge relation, $n$-dependence corresponds to the inability to encode a random $(n+1)$-partite $(n+1)$-hypergraph with a definable edge relation. We characterize $n$-dependence by counting $\varphi$-types over finite sets (generalizing the Sauer–Shelah lemma, answering a question of Shelah), and in terms of the collapse of random ordered $(n+1)$-hypergraph indiscernibles down to order-indiscernibles (which implies that the failure of $n$-dependence is always witnessed by a formula in a single free variable).

#### Article information

Source
Notre Dame J. Formal Logic, Volume 60, Number 2 (2019), 195-214.

Dates
Accepted: 13 December 2016
First available in Project Euclid: 6 May 2019

https://projecteuclid.org/euclid.ndjfl/1557129619

Digital Object Identifier
doi:10.1215/00294527-2019-0002

Mathematical Reviews number (MathSciNet)
MR3952231

Zentralblatt MATH identifier
07096536

#### Citation

Chernikov, Artem; Palacin, Daniel; Takeuchi, Kota. On $n$ -Dependence. Notre Dame J. Formal Logic 60 (2019), no. 2, 195--214. doi:10.1215/00294527-2019-0002. https://projecteuclid.org/euclid.ndjfl/1557129619

#### References

• [1] Abramson, F. G., and L. A. Harrington, “Models without indiscernibles,” Journal of Symbolic Logic, vol. 43 (1978), no. 3, pp. 572–600.
• [2] Adler, H., “An introduction to theories without the independence property,” to appear in Archive for Mathematical Logic, preprint, 2008, http://www.logic.univie.ac.at/~adler/docs/nip.pdf.
• [3] Aschenbrenner, M., A. Dolich, D. Haskell, D. Macpherson, and S. Starchenko, “Vapnik-Chervonenkis density in some theories without the independence property, I,” Transactions of the American Mathematical Society, vol. 368 (2016), no. 8, pp. 5889–949.
• [4] Beyarslan, Ö., “Random hypergraphs in pseudofinite fields,” Journal of the Institute of Mathematics of Jussieu, vol. 9 (2010), no. 1, pp. 29–47.
• [5] Bohman, T., and P. Keevash, “The early evolution of the $H$-free process,” Inventiones Mathematicae, vol. 181 (2010), no. 2, pp. 291–336.
• [6] Bollobás, B., Extremal Graph Theory, Dover Publications, Mineola, New York, 2004.
• [7] Cherlin, G. L., and E. Hrushovski, Finite Structures with Few Types, vol. 152 of Annals of Mathematics Studies, Princeton University Press, Princeton, 2003.
• [8] Chernikov, A., and P. Simon, “Externally definable sets and dependent pairs II,” Transactions of the American Mathematical Society, vol. 367 (2015), no. 7, pp. 5217–35.
• [9] Erdős, P., “On extremal problems of graphs and generalized graphs,” Israel Journal of Mathematics, vol. 2 (1964), pp. 183–90.
• [10] Haskell, D., E. Hrushovski, and D. Macpherson, Stable Domination and Independence in Algebraically Closed Valued Fields, vol. 30 of Lecture Notes in Logic, Association for Symbolic Logic, Chicago, 2008.
• [11] Hempel, N., “On $n$-dependent groups and fields,” Mathematical Logic Quarterly, vol. 62 (2016), no. 3, pp. 215–224.
• [12] Hrushovski, E., “On pseudo-finite dimensions,” Notre Dame Journal of Formal Logic, vol. 54 (2013), nos. 3–4, pp. 463–495.
• [13] Hrushovski, E., and A. Pillay, “On NIP and invariant measures,” J. Eur. Math. Soc. (JEMS), vol. 13 (2011), no. 4, pp. 1005–1061.
• [14] Nešetřil, J., and V. Rödl, “Partitions of finite relational and set systems,” J. Combinatorial Theory Series A, vol. 22 (1977), no. 3, pp. 289–312.
• [15] Nešetřil, J., and V. Rödl, “Ramsey classes of set systems,” Journal of Combinational Theory Series A, vol. 34 (1983), no. 2, pp. 183–201.
• [16] Ngo, H. Q., “Three proofs of Sauer-Shelah lemma,” lecture notes, http://www.cse.buffalo.edu/~hungngo/classes/2010/711/lectures/sauer.pdf.
• [17] Pach, J., and P. K. Agarwal, Combinatorial Geometry, Wiley, New York, 1995.
• [18] Scow, L., “Characterization of NIP theories by ordered graph-indiscernibles,” Annals of Pure and Applied Logic, vol. 163 (2012), no. 11, pp. 1624–41.
• [19] Scow, L., “Indiscernibles, EM-types, and Ramsey classes of trees,” Notre Dame Journal of Formal Logic, vol. 56 (2015), no. 3, pp. 429–47.
• [20] Shelah, S., Classification Theory and the Number of Nonisomorphic Models, 2nd edition, vol. 92 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1990.
• [21] Shelah, S., “Strongly dependent theories,” Israel Journal of Mathematics, vol. 204 (2014), no. 1, pp. 1–83.
• [22] Shelah, S., “Definable groups for dependent and 2-dependent theories,” Sarajevo Journal of Mathematics, vol. 13(25) (2017), no. 1, pp. 3–25.
• [23] Shelah, S., “Dependent dreams: recounting types,” preprint, arXiv:1202.5795 [math.LO].
• [24] Takeuchi, K., “A characterization of $n$-dependent theories,” Lecture Notes: RIMS Kôkyûroku Bessatsu, vol. 1888 (2014), no. 2014.4, pp. 59–66.
• [25] Vapnik, V. N., and A. Y. Chervonenkis, “On the uniform convergence of relative frequencies of events to their probabilities,” Theory of Probability and Its Applications, vol. 16 (1971), no. 2, pp. 264–80.