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SET MAPPINGS WITH FREE SETS WHICH ARE ARITHMETIC PROGRESSIONS
Part of:
Set theory
Published online by Cambridge University Press: 06 February 2018
Abstract
If 
$3\leqslant n<\unicode[STIX]{x1D714}$ and 
$V$ is a vector space over 
$\mathbb{Q}$ with 
$|V|\leqslant \aleph _{n-2}$, then there is a well ordering of 
$V$ such that every vector is the last element of only finitely many length-
$n$ arithmetic progressions (
$n$-APs). This implies that there is a set mapping 
$f:V\rightarrow [V]^{{<}\unicode[STIX]{x1D714}}$ with no free set which is an 
$n$-AP. If, however, 
$|V|\geqslant \aleph _{n-1}$, then for every set mapping 
$f:V\rightarrow [V]^{{<}\unicode[STIX]{x1D714}}$ there is a free set which is an 
$n$-AP.
MSC classification
- Type
- Research Article
- Information
- Copyright
- Copyright © University College London 2018
References
Ceder, J., On decomposition into anticonvex sets. Rev. Roumaine Math. Pures Appl.
14
1969, 955–961.Google Scholar
Erdős, P., Problems and results in chromatic graph theory. In Proof Techniques in Graph Theory, Academic Press (New York, NY, 1969), 27–35.Google Scholar
Erdős, P. and Kakutani, S., On nondenumerable graphs. Bull. Amer. Math. Soc.
49
1943, 457–461.CrossRefGoogle Scholar
Erdős, P. and Komjáth, P., Countable decompositions of ℝ2 and ℝ3
. Discrete Comput. Geom.
5
1990, 325–331.CrossRefGoogle Scholar
Fodor, G., Proof of a conjecture of Erdős. Acta Sci. Math. (Szeged)
14
1951–1952, 219–227.Google Scholar
Hindman, N., Leader, I. and Strauss, D., Pairwise sums in colourings of the reals. Abh. Math. Semin. Univ. Hambg.
87
2017, 275–287, doi:10.1007/s12188-016-0166-x.CrossRefGoogle Scholar
Kuratowski, K., Sur une charactérisation des alephs. Fund. Math.
38
1951, 14–17.CrossRefGoogle Scholar
Sierpiński, W., Sur quelques propositions concernant la puissance du continu. Fund. Math.
38
1951, 1–13.CrossRefGoogle Scholar
Sierpiński, W., Hypothèse du Continu, 2nd edn., Chelsea Publishing Company (New York, 1956).Google Scholar