On the structure of large sum-free sets of integers

A set of integers is called sum-free if it contains no triple $(x,y,z)$ of not necessarily distinct elements with $x+y=z$. In this paper, we provide a structural characterisation of sum-free subsets of $\{1,2,\dots ,n\}$ of density at least $2\mathrm{/}5-c$, where $c$ is an absolute positive constant. As an application, we derive a stability version of Hu’s Theorem [Proc. Amer. Math. Soc. 80 (1980), 711-712] about the maximum size of a union of two sum-free sets in $\{1,2,\dots ,n\}$. We then use this result to show that the number of subsets of $\{1,2,\dots ,n\}$ which can be partitioned into two sum-free sets is $\mathrm{\Theta }(2^{4n\mathrm{/}5})$, confirming a conjecture of Hancock, Staden and Treglown [arXiv:1701.04754].