Generalized sum-free subsets

Caro, Yair

doi:https://doi.org/10.1155/S016117129000103X

International Journal of Mathematics and Mathematical Sciences

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Volume 13 |Article ID 191240 | https://doi.org/10.1155/S016117129000103X

Yair Caro, "Generalized sum-free subsets", International Journal of Mathematics and Mathematical Sciences, vol. 13, Article ID 191240, 4 pages, 1990. https://doi.org/10.1155/S016117129000103X

Generalized sum-free subsets

Yair Caro¹

¹School of Education, University of Haifa - Oranim, Tivon, Israel

Received21 Nov 1989

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.

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Copyright © 1990 Hindawi Publishing Corporation. 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.

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