On the canonical Ramsey theorem of Erd˝os and Rado: a short proof using ultrafilter theory
https://doi.org/10.22405/2226-8383-2024-25-3-396-407
Abstract
The paper gives a short proof of the canonical Ramsey theorem of Erd˝os and Rado using ultrafilter theory.
About the Authors
Nadja Abdul Vahidovna MirRussian Federation
Nikolay Lvovich Polyakov
Russian Federation
candidate of physical and mathematical science
References
1. Ramsey F.P. 1930, “On a problem of formal logic”, Proc. London Math. Soc., Vol. 30, pp.
2. –286.
3. Jeh, T. 2002,“Set theory. The Third Millennium Edition, revised and expanded”, Springer, 769 p.
4. Graham, R. L., Rothschild, B. L., Spencer, J. H., Solymosi, J. 2015, “Ramsey Theory. 3rd ed.”, John Wiley and Sons, NY, 99 p.
5. Erd˝os, P., Rado, R. 1950, “A combinatorial theorem”, J. London Math. Soc., Vol. 25, pp. 249–255.
6. Matet, P. 2016, “An easier proof of the Canonical Ramsey Theorem”, Colloquium Mathematicum, Vol. 145, pp. 187–191.
7. Halbeisen, L. J. 2012, “Combinatorial Set Theory”, Springer, London, 594 p.
8. Erd˝os, P., Rado, R. 1952, “Combinatorial Theorems on Classifications of Subsets of a Given Set”, Proc. London Math. Soc., vol. s3–2, no. 1, pp. 417–439.
9. Rado, R. 1986, “Note on Canonical Partitions”, Bul. of the London Math. Soc., vol. 18, no. 2, pp. 123–126.
10. Mileti, J. R. 2008, “The canonical Ramsey theorem and computability theory”, Trans. Amer. Math. Soc., vol. 360, pp. 1309–1341.
11. Lefmann, H., R¨odl, V. 1995, “On Erd˝os-Rado numbers”, Combinatorica, vol. 15, pp. 85–104.
12. Polyakov, N. L. 2023, “On the Canonical Ramsey Theorem of Erd˝os and Rado and Ramsey Ultrafilters”, Dokl. Math., vol. 108, pp. 392–401.
13. Comfort, W. 1977, “Ultrafilters: Some old and some new results”, Bull. Amer. Math. Soc., vol. 83, pp. 417–455.
14. Di Nasso, M., Goldbring, I., Lupini M,. 2019, “Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory”, Springer, 206 p.
15. Comfort, W. W., Negrepontis, S. 1974, “The theory of ultrafilters”, Springer, Berlin, 481 p.
16. Hindman, N., Strauss, D. 2012, “Algebra in the Stone–ˇCech Compactification. 2nd ed., revised and expanded”, W. de Gruyter, Berlin–N.Y., 591 p.
17. Jeh, T. 1971, “Lectures in Set Theory: With Particular Emphasis on the Method of Forcing”, Springer-Verlag, 148 p.
18. Polyakov, N. L., Shamolin M. V. 2014, “On a generalization of Arrow’s impossibility theorem”, Dokl. Math., vol. 89, pp. 290–292.
19. Goranko, V. 2007, “Filter and ultrafilter extensions of structures: universal-algebraic aspects”, preprint, 30 p.
20. Saveliev, D. I. 2011, “Ultrafilter extensions of models”, Lecture Notes in AICS, vol. 6521, pp. 162–177.
21. Saveliev, D. I. 2012, “On ultrafilter extensions of models”, In: S.-D. Friedman et al. (eds.). The Infinity Project Proc. CRM Documents 11, Barcelona, pp. 599–616.
22. Saveliev, D. I., Shelah, S. 2019, “Ultrafilter extensions do not preserve elementary equivalence”, Math. Log. Quart., vol. 65, pp. 511–516.
23. Saveliev, D. I. 2014, “On idempotents in compact left topological universal algebras”, Topology Proc., vol. 43, pp. 37–46.
24. Poliakov, N. L., Saveliev, D. I. 2017, “On two concepts of ultrafilter extensions of firstorder models and their generalizations”, Logic, Language, Information, and Computation, Lecture Notes in Computer Science, eds. J. Kennedy, R. J. G. B. de Queiroz, Springer, Berlin, Heidelberg, vol. 10388, pp. 336–348.
25. Poliakov, N. L., Saveliev, D. I. 2021, “On ultrafilter extensions of first-order models and
26. ultrafilter interpretations”, Arch. Math. Logic, vol. 60, pp. 625–681.
27. Wimmers, E. 1982, “The Shelah P-point independence theorem”, Israel Journal of Mathematics, vol. 43, no. 1, pp. 28–48.
Review
For citations:
Mir N.A., Polyakov N.L. On the canonical Ramsey theorem of Erd˝os and Rado: a short proof using ultrafilter theory. Chebyshevskii Sbornik. 2024;25(3):396-407. (In Russ.) https://doi.org/10.22405/2226-8383-2024-25-3-396-407