On strongly sum-free subsets of abelian groups

Volume 71 / 1996

Tomasz Łuczak, Tomasz Schoen Colloquium Mathematicum 71 (1996), 149-151 DOI: 10.4064/cm-71-1-149-151


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.


  • Tomasz Łuczak
  • Tomasz Schoen

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