Generalized sum-free subsets
Given a finite set \(S\) of non-zero real numbers and a finite set \(A\) of positive integers, denote \(S^ A=\{\sum^{a}_{i=1}s_ i\); \(a\in A\), \(s_ i\in S\}\), where in the sum the \(s_ i\)'s are not necessarily distinct. If \(F=\{A(i): 1\leq i\leq t\}\) is a finite collection of pairwise disjoint subsets of positive integers, then the author calls \(S\) to be \(F\)-free if for every \(A(i), A(j)\) with \(i\neq j\) we have \(S^{A(i)}\cap S^{A(j)}=\emptyset\). This generalizes the notion of the sum-free subsets (where \(F=\{\{1\},\{2\}\})\). Firstly, the author gives an ingenious combinatorial proof of a special case of a recent result of \textit{N. Alon} and \textit{D. J. Kleitman} [A tribute to Paul Erdős, 13--26 (1990; Zbl 0718.11006)], namely that any finite set \(B\) of positive integers contains a sum-free subset \(A\) of cardinality \(| A| >| B| /3\). Then he proves (however, the proof is not clear at every step) that if \(S\) and \(F\) are as above and \(t\geq 2\) then \(S\) contains an \(F\)-free subset \(Q\) such that \(| Q| \geq c(F)| S|\), where \(c(F)\) is a positive constant depending only on \(F\). Applications to solutions of linear equations over integers are also given.
scientific article; zbMATH DE number 4208148