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Transversals for strongly almost disjoint families

Author: Paul J. Szeptycki
Journal: Proc. Amer. Math. Soc. 135 (2007), 2273-2282
MSC (2000): Primary 03E05; Secondary 03E50
Published electronically: February 28, 2007
MathSciNet review: 2299505
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Abstract: For a family of sets $ A$, and a set $ X$, $ X$ is said to be a transversal of $ A$ if $ X\subseteq \bigcup A$ and $ \vert a\cap X\vert=1$ for each $ a\in A$. $ X$ is said to be a Bernstein set for $ A$ if $ \emptyset\not=a\cap X\not=a$ for each $ a\in A$. Erdos and Hajnal first studied when an almost disjoint family admits a set such as a transversal or Bernstein set. In this note we introduce the following notion: a family of sets $ A$ is said to admit a $ \sigma$-transversal if $ A$ can be written as $ A=\bigcup\{A_n:n\in \omega\}$ such that each $ A_n$ admits a transversal. We study the question of when an almost disjoint family admits a $ \sigma$-transversal and related questions.

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Additional Information

Paul J. Szeptycki
Affiliation: Department of Mathematics and Statistics, York University, Toronto, OntarioCanada M3J 1P3

Keywords: Almost disjoint family, transversal, Bernstein partition
Received by editor(s): November 23, 2005
Received by editor(s) in revised form: December 8, 2005, and March 1, 2006
Published electronically: February 28, 2007
Additional Notes: The author acknowledges support from NSERC grant 238944
Communicated by: Julia Knight
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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