No CrossRef data available.
Article contents
SET MAPPINGS WITH FREE SETS WHICH ARE ARITHMETIC PROGRESSIONS
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