This posts gives some results for the sum-free subset problem in the previous post.
Paul Erdős proved in 1965 that a set of n non-zero integers contains a sum-free subset of size larger than kn where k = 1/3.
The best value of k is unknown. Erdős proved k is at least 1/3. Alon and Kleitman proved that kmust be less than 12/29.
Alon and Kleitman also consider the problem of sum-free subsets of an arbitrary Abelian group. They proved that set of n non-zero elements of an Abelian group must have a sum-free subset of at least 2n/7 elements. The constant 2/7 is the best value in general. But for the specific case of the integers one can do better, since 1/3 > 2/7, but the best constant for the integer case is unknown.
Source: The Probabilistic Method