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A subset $A$ of a given finite abelian group $G$ is called $(k,l)$-sum-freeif the sum of $k$ (not necessarily distinct) elements of $A$ does not equal thesum of $l$ (not necessarily distinct) elements of $A$. We are interested infinding the maximum size $\lambda_{k,l}(G)$ of a $(k,l)$-sum-free subset in$G$.  A $(2,1)$-sum-free set is simply called a sum-free set. The maximum size of asum-free set in the cyclic group $\mathbb{Z}_n$ was found almost forty yearsago by Diamanda and Yap; the general case for arbitrary finite abelian groupswas recently settled by Green and Ruzsa. Here we find the value of$\lambda_{3,1}(\mathbb{Z}_n)$. More generally, a recent paper of Hamidoune andPlagne examines $(k,l)$-sum-free sets in $G$ when $k-l$ and the order of $G$are relatively prime; we extend their results to see what happens without thisassumption.

ON THE MAXIMUM SIZE OF A (k,l)-SUM-FREE SUBSET OF AN ABELIAN GROUP