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SET MAPPINGS WITH FREE SETS WHICH ARE ARITHMETIC PROGRESSIONS

  • Péter Komjáth (a1)
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.

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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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