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RAMSEY’S COHEIRS

Part of: Nonstandard models Model theory Extremal combinatorics

Published online by Cambridge University Press:  15 February 2021

EUGENIO COLLA  and
DOMENICO ZAMBELLA
EUGENIO COLLA
Affiliation:
DIPARTIMENTO DI MATEMATICA UNIVERSITÀ DEGLI STUDI DI TORINOTORINO10100, ITALYE-mail: eugenio.colla@unito.itE-mail: domenico.zambella@unito.it
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Abstract

We use the model theoretic notion of coheir to give short proofs of old and new theorems in Ramsey Theory. As an illustration we start from Ramsey’s theorem itself. Then we prove Hindman’s theorem and the Hales–Jewett theorem. Finally, we prove two Ramsey theoretic principles that have among their consequences partition theorems due to Carlson and to Gowers.

Keywords

model theoretic methodsRamsey theoryHales-Jewett theorempartition regularity

MSC classification

Primary: 05D10: Ramsey theory
Secondary: 03C98: Applications of model theory 03H99: None of the above, but in this section
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 1 , March 2022 , pp. 377 - 391
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Agostini, C. and Colla, E., Ramsey monoids. Preprint, 2020. arXiv:2012.02506.Google Scholar
Andrews, U., Conant, G., and Goldbring, I., Definable sets containing productsets in expansions of groups. Journal of Group Theory , vol. 22 (2019), pp. 63–86.10.1515/jgth-2018-0038CrossRefGoogle Scholar
Bergelson, V., Blass, A., and Hindman, N., Partition theorems for spaces of variable words . Proceedings of the London Mathematical Society (3) , vol. 68 (1994), no. 3, pp. 449–476.10.1112/plms/s3-68.3.449CrossRefGoogle Scholar
Blass, A., Ultrafilters: Where topological dynamics $=$ algebra $=$ combinatorics . Topology Proceedings , vol. 18 (1993), pp. 33–56.Google Scholar
Di Nasso, M., Hypernatural numbers as ultrafilters , Nonstandard Analysis for the Working Mathematician (P. A. Loeb and M. P. H. Wolff, editors), Springer, 2015, pp. 443–474.10.1007/978-94-017-7327-0_11CrossRefGoogle Scholar
Di Nasso, M., Goldbring, I., and Lupini, M., Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory , Lecture Notes in Mathematics, vol. 2239, Springer, 2019.Google Scholar
Dodos, P. and Kanellopoulos, V., Ramsey Theory for Product Spaces , Mathematical Surveys and Monographs, vol. 212, American Mathematical Society, Providence, RI, 2016.10.1090/surv/212CrossRefGoogle Scholar
Gowers, W. T., Lipschitz functions on classical spaces . European Journal of Combinatorics , vol. 13 (1992), no. 3, pp. 141–151.10.1016/0195-6698(92)90020-ZCrossRefGoogle Scholar
Hales, A. W. and Jewett, R. I., Regularity and positional games . Transactions of the American Mathematical Society , vol. 106 (1963), pp. 222–229.CrossRefGoogle Scholar
Hindman, N., Finite sums from sequences within cells of a partition of N . Journal of Combinatorial Theory Series A , vol. 17, (1974), pp. 1–11.10.1016/0097-3165(74)90023-5CrossRefGoogle Scholar
Koppelberg, S., The Hales–Jewett theorem via retractions, Topology Proceedings , vol. 28, 2004, pp. 595–601.Google Scholar
Lupini, M., Actions on semigroups and an infinitary Gowers–Hales–Jewett Ramsey theorem . Transactions of the American Mathematical Society , vol. 371 (2019), pp. 3083–3116.10.1090/tran/7337CrossRefGoogle Scholar
Milliken, K. R., Ramsey’s theorem with sums or unions . Journal of Combinatorial Theory Series A , vol. 18 (1975), pp. 276–290.10.1016/0097-3165(75)90039-4CrossRefGoogle Scholar
Newelski, L., Topological dynamics of definable group actions , this Journal, vol. 74 (2009), no. 1, pp. 50–72.Google Scholar
Newelski, L., Topological dynamics of stable groups , this Journal, vol. 79 (2014), no. 4, pp. 1199–1223.Google Scholar
Shelah, S., Primitive recursive bounds for van der Waerden numbers . Journal of the American Mathematical Society , vol. 1 (1988), no. 3, pp. 683–697.CrossRefGoogle Scholar
Taylor, A. D., A canonical partition relation for finite subsets of ω . Journal of Combinatorial Theory Series A , vol. 21 (1976), no. 2, pp. 137–146.10.1016/0097-3165(76)90058-3CrossRefGoogle Scholar
Tent, K. and Ziegler, M., A Course in Model Theory , Lecture Notes in Logic, vol. 40, Association for Symbolic Logic, Cambridge University Press, Cambridge, 2012.CrossRefGoogle Scholar
Zambella, D., A crèche course in model theory, AMS Open Math Notes, 2018.Google Scholar