If $k$ is a positive integer, we say that a set $A$ of positive integers is $k$-sum-free if there do not exist $a,b,c$ in $A$ such that $a + b = kc$. In particular we give a precise characterization of the structure of maximum sized $k$-sum-free sets in $\{1,\ldots,n\}$ for $k\ge 4$ and $n$ large.

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An (N, K, ) connector (rearrangeable network) is a digraph with N inputs and K outputs such that for any injective mapping from the outputs to the inputs there is a set of vertex disjoint paths of length implementing it. The study of this problem begins with the pioneering work of Shannon, Slepian, Clos and Beneš in the 1950's who studied the balanced N = K case, and arbitrary depth , and later on by Pippenger and Yao who used expanders to construct small depth, balanced connectors.In the last decade there has been research on the unbalanced case. This started with Oruç who showed an explicit construction with O(N ) edges, for K ≤ N log(N ) , and proved some lower bounds on small depth networks. Later on Baltz, Jäger, and Srivastav suggested an explicit depth two construction for the K ≤ √ N case with O(N √ K) edges, which was followed by an entirely different depth two construction by Ahlswede and Aydinian with O(N log K) edges, but which worked only for K ≤ N O(1/ √ log N ) . We show that non-constructively depth two connectors with O(N ) edges exist for K ≤ N 1/2− , for every constant > 0. Furthermore, for that range we show an explicit construction with only N poly(log N ) edges.The solution is based on new expander techniques that were developed in the study of extractors. We believe that these new tools can be very useful in other network constructions as well. We feel that despite the success of these newer tools in theoretical computer science they are still widely neglected in network design, partly because they are so new. We hope our work will find other applications in network design.