Partitioning triples and partially ordered sets
HTML articles powered by AMS MathViewer
- by Albin L. Jones PDF
- Proc. Amer. Math. Soc. 136 (2008), 1823-1830
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. Erdős 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
- J. Baumgartner and A. Hajnal, A proof (involving Martin’s axiom) of a partition relation, Fund. Math. 78 (1973), no. 3, 193–203. MR 319768, DOI 10.4064/fm-78-3-193-203
- P. Erdös and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427–489. MR 81864, DOI 10.1090/S0002-9904-1956-10036-0
- Thomas Jech, Set theory, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1997. MR 1492987, DOI 10.1007/978-3-662-22400-7
- A. L. Jones, Some results in the partition calculus, Ph.D. thesis, Dartmouth College, June 1999.
- Albin L. Jones, A short proof of a partition relation for triples, Electron. J. Combin. 7 (2000), Research Paper 24, 9. MR 1755613
- Albin L. Jones, More on partitioning triples of countable ordinals, Proc. Amer. Math. Soc. 135 (2007), no. 4, 1197–1204. MR 2262926, DOI 10.1090/S0002-9939-06-08538-8
- E. C. Milner and K. Prikry, A partition theorem for triples, Proc. Amer. Math. Soc. 97 (1986), no. 3, 488–494. MR 840635, DOI 10.1090/S0002-9939-1986-0840635-8
- E. C. Milner and K. Prikry, A partition relation for triples using a model of Todorčević, Discrete Math. 95 (1991), no. 1-3, 183–191. Directions in infinite graph theory and combinatorics (Cambridge, 1989). MR 1141938, DOI 10.1016/0012-365X(91)90336-Z
- F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264–286.
- Stevo Todorčević, Partition relations for partially ordered sets, Acta Math. 155 (1985), no. 1-2, 1–25. MR 793235, DOI 10.1007/BF02392535
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
- MR Author ID: 662270
- Email: alj@mojumi.net
- 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
- © Copyright 2007 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
- MathSciNet review: 2373614