Skip to main content Accessibility help


  • Péter Komjáth (a1)

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.

Hide All
1. Ceder, J., On decomposition into anticonvex sets. Rev. Roumaine Math. Pures Appl. 14 1969, 955961.
2. Erdős, P., Problems and results in chromatic graph theory. In Proof Techniques in Graph Theory, Academic Press (New York, NY, 1969), 2735.
3. Erdős, P. and Kakutani, S., On nondenumerable graphs. Bull. Amer. Math. Soc. 49 1943, 457461.
4. Erdős, P. and Komjáth, P., Countable decompositions of ℝ2 and ℝ3 . Discrete Comput. Geom. 5 1990, 325331.
5. Fodor, G., Proof of a conjecture of Erdős. Acta Sci. Math. (Szeged) 14 1951–1952, 219227.
6. Hindman, N., Leader, I. and Strauss, D., Pairwise sums in colourings of the reals. Abh. Math. Semin. Univ. Hambg. 87 2017, 275287, doi:10.1007/s12188-016-0166-x.
7. Kuratowski, K., Sur une charactérisation des alephs. Fund. Math. 38 1951, 1417.
8. Sierpiński, W., Sur quelques propositions concernant la puissance du continu. Fund. Math. 38 1951, 113.
9. Sierpiński, W., Hypothèse du Continu, 2nd edn., Chelsea Publishing Company (New York, 1956).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed