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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

Péter Komjáth
Péter Komjáth*
Affiliation:
Institute of Mathematics, Eötvös University, Budapest, Pázmány P. s. 1/C 1117, Hungary email [email protected]
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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

Primary: 03E05: Other combinatorial set theory
Type
Research Article
Information
Mathematika , Volume 64 , Issue 1 , 2018 , pp. 71 - 76
Copyright
Copyright © University College London 2018 

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