Counting sets of integers, no \(k\) of which sum to another (Q1912293)

A set \(S\subseteq N\) is said to be sum-free if \(S\) contains no \(x,y,z\) (not necessarily distinct) such that \(x+y =z\). Erdös and Granville have shown that the number of all sum-free subsets of \(\{1,2,\dots,n\}\) is \(o(2^{n({1 \over 2} + \varepsilon)})\) for every \(\varepsilon> 0\). Erdös has asked whether the number of all subsets of \(\{1,2, \dots,n\}\) without a solution of \(x+y+z=t\) is \(c\cdot 2^{2n \over 3}\). In this paper the authors give the answer to this question in the affirmative and show a more general result according to which the number of all subsets of \(\{1,2, \dots, n\}\) with no solution of \(x_1+ \cdots +x_k=y\) is at most \(c \cdot 2^{\alpha n}\), where \(\alpha= {k-1 \over k} (k\geq 3)\).