EUDML

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 $n \geq n_{0}$ 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 $a_{i_{j}} a_{i_{k}} \neq a_{m}$ for $1 \leq j < k \leq l$ and $1 \leq m \leq n$ . 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.

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A note on the number of $(k, l)$ -sum-free sets.

Multiple set addition in $ℤ_{p}$ .

Integer sets having the maximum number of distinct differences.

On sets of natural numbers without solution to a noninvariant linear equation

Arithmetic progressions in sums of subsets of sparse sets

The number of (2,3)-sum-free subsets of {1,...,n}

The structure of maximum subsets of ${1, \dots, n}$ with no solutions to $a + b = k c$ .

Small bases for finite groups

On strongly sum-free subsets of abelian groups

On the maximal density of sum-free sets

The number of solutions of a homogeneous linear congruence

On sumsets and spectral gaps

On a problem of Konyagin