Yair Caro

Copyright © 1990 Yair Caro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let F={A(i):1≤i≤t, t≥2}, be a finite collection of finite, pairwise disjoint subsets of Z+. Let S⊂R\{0} and A⊂Z+ be finite sets. Denote by SA={∑i=1asi:a∈A, Si∈S, the si are not necessarily distinct}. For S and F as above we say that S is F-free if for every A(i), A(j)∈F, i≠j, SA(i)⋂SA(j)=ϕ.

We prove that for S and F as above, S contains an F-free subset Q such that |Q|≥c(F)|S|, when c(F) is a positive constant depending only on F.

This result generalizes earlier results of Erdos [3] and Alon and Kleitman [2], on sum-free subsets. Several possible extensions are also discussed.