Convexity numbers of closed sets in
Authors:
Stefan Geschke and Menachem Kojman
Journal:
Proc. Amer. Math. Soc. 130 (2002), 28712881
MSC (1991):
Primary 05A20, 52A05, 03E17, 03E35; Secondary 03E75
Published electronically:
March 25, 2002
MathSciNet review:
1908910
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Abstract 
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Abstract: For let be the ideal in generated by all sets which do not contain equidistant points in the usual metric on . For each a set is constructed in so that the ideal which is generated by the convex subsets of restricted to the convexity radical is isomorphic to . Thus is equal to the least number of convex subsets required to cover  the convexity number of . For every nonincreasing function we construct a model of set theory in which for each . When is strictly decreasing up to , uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of . It is conjectured that , but never more than , different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of . This conjecture is true for and .
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A. Blass, Simple cardinal characteristics of the continuum. Set theory of the reals (Ramat Gan, 1991), 6390, Israel Math. Conf. Proc., 6, BarIlan Univ., Ramat Gan, 1993. MR 94i:03098
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S. Geschke, M. Kojman, W. Kubis, R. Schipperus, Convex decompositions in the plane and continuous pair colorings of the irrationals, submitted.
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L. Newelski, A. Roslanowski, The ideal determined by the unsymmetric game, Proc. Amer. Math. Soc. 117 (1993), no. 3, 823831. MR 93d:03053
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S. Shelah, J. Steprans, The covering numbers of Mycielski ideals are all equal, J. Symbolic Logic 66 (2001), 707718.
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J. Steprans, Decomposing Euclidean space with a small number of smooth sets., Trans. Amer. Math. Soc. 351 (1999), no. 4, 14611480. MR 99f:04002
 9.
J. Zapletal, Isolating cardinal invariants, preprint.
 1.
 T. Bartoszynski, H. Judah, Set Theory: On the structure of the real line. A K Peters, Ltd., Wellesley, MA, 1995. MR 96k:03002
 2.
 A. Blass, Simple cardinal characteristics of the continuum. Set theory of the reals (Ramat Gan, 1991), 6390, Israel Math. Conf. Proc., 6, BarIlan Univ., Ramat Gan, 1993. MR 94i:03098
 3.
 S. Geschke, M. Kojman, W. Kubis, R. Schipperus, Convex decompositions in the plane and continuous pair colorings of the irrationals, submitted.
 4.
 M. Goldstern and S. Shelah, Many simple cardinal invariants of the continuum, Arch. Math. Logic 32 (1993), no. 3, 203221. MR 94c:03064
 5.
 T. Jech, Multiple Forcing, Cambridge Tracts in Mathematics, 88. Cambridge University Press, Cambridge, 1986. MR 89h:03001
 6.
 L. Newelski, A. Roslanowski, The ideal determined by the unsymmetric game, Proc. Amer. Math. Soc. 117 (1993), no. 3, 823831. MR 93d:03053
 7.
 S. Shelah, J. Steprans, The covering numbers of Mycielski ideals are all equal, J. Symbolic Logic 66 (2001), 707718.
 8.
 J. Steprans, Decomposing Euclidean space with a small number of smooth sets., Trans. Amer. Math. Soc. 351 (1999), no. 4, 14611480. MR 99f:04002
 9.
 J. Zapletal, Isolating cardinal invariants, preprint.
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Additional Information
Stefan Geschke
Affiliation:
Freie Universität Berlin, Arnimallee 26, D1495 Berlin
Email:
geschke@math.fuberlin.de
Menachem Kojman
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel
Email:
kojman@math.bgu.ac.il
DOI:
http://dx.doi.org/10.1090/S0002993902064377
PII:
S 00029939(02)064377
Keywords:
Convex cover,
convexity number,
$n$space,
forcing extension,
covering number
Received by editor(s):
April 19, 2001
Received by editor(s) in revised form:
May 31, 2001
Published electronically:
March 25, 2002
Communicated by:
Carl G. Jockusch, Jr.
Article copyright:
© Copyright 2002 American Mathematical Society
