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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Perfect cliques and $G_\delta$ colorings of Polish spaces


Author: Wieslaw Kubis
Journal: Proc. Amer. Math. Soc. 131 (2003), 619-623
MSC (2000): Primary 52A37, 54H05; Secondary 03E02, 52A10
Published electronically: August 19, 2002
MathSciNet review: 1933354
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Abstract: A coloring of a set $X$ is any subset $C$ of $[X]^N$, where $N>1$ is a natural number. We give some sufficient conditions for the existence of a perfect $C$-homogeneous set, in the case where $C$ is $G_\delta$ and $X$ is a Polish space. In particular, we show that it is sufficient that there exist $C$-homogeneous sets of arbitrarily large countable Cantor-Bendixson rank. We apply our methods to show that an analytic subset of the plane contains a perfect $3$-clique if it contains any uncountable $k$-clique, where $k$ is a natural number or $\aleph_0$ (a set $K$ is a $k$-clique in $X$ if the convex hull of any of its $k$-element subsets is not contained in $X$).


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Additional Information

Wieslaw Kubis
Affiliation: Department of Mathematics, University of Silesia, Katowice, Poland
Address at time of publication: Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Email: kubis@math.bgu.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06584-X
PII: S 0002-9939(02)06584-X
Keywords: Open ($G_\delta$) coloring, perfect homogeneous set, clique
Received by editor(s): August 20, 2001
Received by editor(s) in revised form: October 1, 2001
Published electronically: August 19, 2002
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2002 American Mathematical Society