# On strongly sum-free subsets of abelian groups

Colloquium Mathematicae (1996)

• Volume: 71, Issue: 1, page 149-151
• ISSN: 0010-1354

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## Abstract

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In his book on unsolved problems in number theory [1] R. K. Guy asks whether for every natural l there exists ${n}_{0}={n}_{0}\left(l\right)$ with the following property: for every $n\ge {n}_{0}$ and any n elements ${a}_{1},...,{a}_{n}$ of a group such that the product of any two of them is different from the unit element of the group, there exist l of the ${a}_{i}$ such that ${a}_{{i}_{j}}{a}_{{i}_{k}}\ne {a}_{m}$ for $1\le j and $1\le m\le n$. In this note we answer this question in the affirmative in the first non-trivial case when l=3 and the group is abelian, proving the following result.

## How to cite

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Łuczak, Tomasz, and Schoen, Tomasz. "On strongly sum-free subsets of abelian groups." Colloquium Mathematicae 71.1 (1996): 149-151. <http://eudml.org/doc/210420>.

@article{Łuczak1996,
abstract = {In his book on unsolved problems in number theory [1] R. K. Guy asks whether for every natural l there exists $n_0 = n_0(l)$ with the following property: for every $n ≥ n_0$ and any n elements $a_1,...,a_n$ of a group such that the product of any two of them is different from the unit element of the group, there exist l of the $a_i$ such that $a_\{i_j\}a_\{i_k\} ≠ a_m$ for $1 ≤ j < k ≤ l$ and $1 ≤ m ≤ n$. In this note we answer this question in the affirmative in the first non-trivial case when l=3 and the group is abelian, proving the following result.},
author = {Łuczak, Tomasz, Schoen, Tomasz},
journal = {Colloquium Mathematicae},
keywords = {sum-free subsets; abelian group},
language = {eng},
number = {1},
pages = {149-151},
title = {On strongly sum-free subsets of abelian groups},
url = {http://eudml.org/doc/210420},
volume = {71},
year = {1996},
}

TY - JOUR
AU - Łuczak, Tomasz
AU - Schoen, Tomasz
TI - On strongly sum-free subsets of abelian groups
JO - Colloquium Mathematicae
PY - 1996
VL - 71
IS - 1
SP - 149
EP - 151
AB - In his book on unsolved problems in number theory [1] R. K. Guy asks whether for every natural l there exists $n_0 = n_0(l)$ with the following property: for every $n ≥ n_0$ and any n elements $a_1,...,a_n$ of a group such that the product of any two of them is different from the unit element of the group, there exist l of the $a_i$ such that $a_{i_j}a_{i_k} ≠ a_m$ for $1 ≤ j < k ≤ l$ and $1 ≤ m ≤ n$. In this note we answer this question in the affirmative in the first non-trivial case when l=3 and the group is abelian, proving the following result.
LA - eng
KW - sum-free subsets; abelian group
UR - http://eudml.org/doc/210420
ER -

## References

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1. [1] R. K. Guy, Unsolved Problems in Number Theory, Springer, New York, 1994, Problem C14. Zbl0805.11001

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