# Sum-Free Subset Results

# Sum-Free Subset Results

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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 *k*must 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 2*n*/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

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