On the density of sets containing no three distinct numbers with all their sums (Q1578466)

Let \(A\) be a subset of positive integers and let \(X\subseteq A.\) Denote by \(P(X)\) the subset sums of \(X,\) i.e. let \(P(X)=\{\sum_{y\in Y}y : Y\subseteq X, Y\neq \emptyset\}.\) Choi, Erdős and Szemerédi proved that for every \(k\in \mathbb N,\) there exists \(\alpha_k,\) \(0<\alpha_k<1,\) such that if \(\bar d(A)>1-\alpha_k\) then there exists an \(X\subseteq A;\) \(|X|=k\) and \(P(X)\subseteq A.\) It is well-known (an easy exercise) that \(\alpha_2=1/2.\) In the present paper the author investigates the first nontrivial case proving \(\alpha_3=1/3.\) The proof is combinatorial and uses a result of Luczak and Schoen on sum-free set \(A'\) with \(\bar d(A')>1/3.\)