International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 4, Pages 751-754
doi:10.1155/S016117129000103X

Generalized sum-free subsets

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.