On the maximum number of integer colourings with forbidden monochromatic sums

Let $f(n,r)$ denote the maximum number of colourings of $A subseteq lbrace 1,ldots,nrbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $lbrace x,y,zrbrace$ such that $x+y=z$. We show that $f(n,2) = 2^{lceil n/2rceil}$, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of $f(n,r)$ for $r leq 5$. Similar results were obtained by H`an and Jimenez in the setting of finite abelian groups.