Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

   
 
 

 

Partitioning triples and partially ordered sets


Author: Albin L. Jones
Journal: Proc. Amer. Math. Soc. 136 (2008), 1823-1830
MSC (2000): Primary 03E05, 05D10; Secondary 05A18
DOI: https://doi.org/10.1090/S0002-9939-07-09170-8
Published electronically: November 6, 2007
MathSciNet review: 2373614
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that if $ P$ is a partial order and $ P \to (\omega)^1_\omega$, then

(a)
$ P \to (\omega + \omega + 1, 4)^3$, and
(b)
$ P \to (\omega + m, n)^3$ for each $ m, n < \omega$.
Together these results represent the best progress known to us on the following question of P. Erdos and others. If $ P \to (\omega)^1_\omega$, then does $ P \to (\alpha, n)^3$ for each $ \alpha < \omega_1$ and each $ n < \omega$?


References [Enhancements On Off] (What's this?)

  • 1. J. Baumgartner and A. Hajnal, A proof (involving Martin's axiom) of a partition relation, Fund. Math. 78 (1973), no. 3, 193-203. MR 0319768 (47:8310)
  • 2. P. Erdos and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427-489. MR 0081864 (18:458a)
  • 3. Thomas Jech, Set theory, second ed., Springer-Verlag, Berlin, 1997. MR 99b:03061
  • 4. A. L. Jones, Some results in the partition calculus, Ph.D. thesis, Dartmouth College, June 1999.
  • 5. -, A short proof of a partition relation for triples, Electronic Journal of Combinatorics 7(1) (2000), 1-9. MR 1755613 (2001a:03097)
  • 6. -, More on partitioning triples of countable ordinals, Proceedings of the American Mathematical Society (2007). MR 2262926 (2007i:03054)
  • 7. E. C. Milner and K. Prikry, A partition theorem for triples, Proc. Amer. Math. Soc. 97 (1986), no. 3, 488-494. MR 840635 (87f:04006)
  • 8. -, A partition relation for triples using a model of Todorcevic, Discrete Math. 95 (1991), nos. 1-3, 183-191, Directions in infinite graph theory and combinatorics (Cambridge, 1989). MR 1141938 (93i:03065)
  • 9. F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264-286.
  • 10. S. Todorcevic, Partition relations for partially ordered sets, Acta Math. 155 (1985), nos. 1-2, 1-25. MR 793235 (87d:03126)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03E05, 05D10, 05A18

Retrieve articles in all journals with MSC (2000): 03E05, 05D10, 05A18


Additional Information

Albin L. Jones
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045-2142
Address at time of publication: 2153 Oakdale Rd., Pasadena, Maryland 21122
Email: alj@mojumi.net

DOI: https://doi.org/10.1090/S0002-9939-07-09170-8
Keywords: Countable ordinals, non-special tree, partial order, Ramsey theory, triples
Received by editor(s): September 18, 2006
Received by editor(s) in revised form: March 20, 2007
Published electronically: November 6, 2007
Additional Notes: The author would like to thank the University of Kansas for its support of this research.
Communicated by: Julia Knight
Article copyright: © Copyright 2007 Albin L. Jones

American Mathematical Society