Vector sets with no repeated differences

Péter Komjáth

Colloquium Mathematicae (1993)

  • Volume: 64, Issue: 1, page 129-134
  • ISSN: 0010-1354

Abstract

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We consider the question when a set in a vector space over the rationals, with no differences occurring more than twice, is the union of countably many sets, none containing a difference twice. The answer is “yes” if the set is of size at most ℵ 2 , “not” if the set is allowed to be of size ( 2 2 ℵ 0 ) + . It is consistent that the continuum is large, but the statement still holds for every set smaller than continuum.

How to cite

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Komjáth, Péter. "Vector sets with no repeated differences." Colloquium Mathematicae 64.1 (1993): 129-134. <http://eudml.org/doc/210162>.

@article{Komjáth1993,
abstract = {We consider the question when a set in a vector space over the rationals, with no differences occurring more than twice, is the union of countably many sets, none containing a difference twice. The answer is “yes” if the set is of size at most $ℵ_2$, “not” if the set is allowed to be of size $(2^\{2^\{ℵ_0\}\})^\{+\}$. It is consistent that the continuum is large, but the statement still holds for every set smaller than continuum.},
author = {Komjáth, Péter},
journal = {Colloquium Mathematicae},
keywords = {combinatorial set theory; forcing; consistency},
language = {eng},
number = {1},
pages = {129-134},
title = {Vector sets with no repeated differences},
url = {http://eudml.org/doc/210162},
volume = {64},
year = {1993},
}

TY - JOUR
AU - Komjáth, Péter
TI - Vector sets with no repeated differences
JO - Colloquium Mathematicae
PY - 1993
VL - 64
IS - 1
SP - 129
EP - 134
AB - We consider the question when a set in a vector space over the rationals, with no differences occurring more than twice, is the union of countably many sets, none containing a difference twice. The answer is “yes” if the set is of size at most $ℵ_2$, “not” if the set is allowed to be of size $(2^{2^{ℵ_0}})^{+}$. It is consistent that the continuum is large, but the statement still holds for every set smaller than continuum.
LA - eng
KW - combinatorial set theory; forcing; consistency
UR - http://eudml.org/doc/210162
ER -

References

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  1. [1] P. Erdős, Set theoretic, measure theoretic, combinatorial, and number theoretic problems concerning point sets in Euclidean space, Real Anal. Exchange 4 (1978-79), 113-138. Zbl0418.04002
  2. [2] P. Erdős, Some applications of Ramsey's theorem to additive number theory, European J. Combin. 1 (1980), 43-46. Zbl0442.10037
  3. [3] P. Erdős and S. Kakutani, On non-denumerable graphs, Bull. Amer. Math. Soc. 49 (1943), 457-461. Zbl0063.01275
  4. [4] P. Erdős and R. Rado, A partition calculus in set theory, ibid. 62 (1956), 427-489. Zbl0071.05105
  5. [5] D. Fremlin, Consequences of Martin's Axiom, Cambridge University Press, 1984. Zbl0551.03033

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