The number of \((2,3)\)-sum-free subsets of \(\{1,\dots,n\}\) (Q2717623)

A subset \(A\) of positive integers is said to be \((k,m)\)-sum-free if there are no solutions to the equation \(x_1+x_2+\dots +x_k=y_1+y_2+\dots +y_m.\) Denote by \(SF_k(n)\) the family of all \((k+1,k)\)-sum-free subsets of \(\{1,\dots ,n\}.\) A celebrated conjecture of Cameron and Erdős states that \(|SF_1(n)|=O(2^{n/2}).\) NEWLINENEWLINENEWLINEIn the present paper the author proves that NEWLINE\[NEWLINE|SF_2(n)|=2^{\lceil n/2\rceil}+O(2^{n/2-c}),NEWLINE\]NEWLINE where \(c\) is an absolute positive constant. Furthermore the author improves a result of \textit{Yu. Bilu} [Combinatorica 18, 449-459 (1998; Zbl 0968.11017)] proving if \(\varepsilon>0,\) then the number of sum-free subsets \(A\subseteq \{1,\dots ,n\}\) such that either \(|A|<(1/4-\varepsilon)n\) or \(|A|>(1/4+\varepsilon)n\) is \(O_\varepsilon(2^{n/2-\varepsilon^2n}).\)