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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Perfect cliques and $G_\delta$ colorings of Polish spaces
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by Wiesław Kubiś PDF
Proc. Amer. Math. Soc. 131 (2003), 619-623 Request permission


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
  • Wiesław Kubiś
  • 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:
  • 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.
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 619-623
  • MSC (2000): Primary 52A37, 54H05; Secondary 03E02, 52A10
  • DOI:
  • MathSciNet review: 1933354