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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Partitioning $ \alpha$-large sets: Some lower bounds


Authors: Teresa Bigorajska and Henryk Kotlarski
Journal: Trans. Amer. Math. Soc. 358 (2006), 4981-5001
MSC (2000): Primary 05A18
Published electronically: June 15, 2006
MathSciNet review: 2231881
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Abstract: Let $ \alpha\rightarrow(\beta)_m^r$ denote the property: if $ A$ is an $ \alpha$-large set of natural numbers and $ [A]^r$ is partitioned into $ m$ parts, then there exists a $ \beta$-large subset of $ A$ which is homogeneous for this partition. Here the notion of largeness is in the sense of the so-called Hardy hierarchy. We give a lower bound for $ \alpha$ in terms of $ \beta,m,r$ for some specific $ \beta$.


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Additional Information

Teresa Bigorajska
Affiliation: Faculty of Mathematics, Cardinal Stefan Wyszyński University, ul. Dewajtis 5, 01–815 Warszawa, Poland
Email: tebigo@op.pl

Henryk Kotlarski
Affiliation: Faculty of Mathematics, Cardinal Stefan Wyszyński University, ul. Dewajtis 5, 01–815 Warszawa, Poland – and – Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, P.O. Box 137, 00–950 Warszawa, Poland
Email: hkl@impan.gov.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03883-9
PII: S 0002-9947(06)03883-9
Keywords: Ramsey theorem, Hardy hierarchy, $\alpha$--large sets
Received by editor(s): April 26, 2004
Received by editor(s) in revised form: October 18, 2004
Published electronically: June 15, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.