It contains five different sum-free subsets, such as {1} and {2, 3}. If you have a set with a million integers, how big is its biggest sum-free subset? His proof dealt with averages: He found a collection of sum-free subsets and calculated that their average size was N/3. But in such a collection, the biggest subsets are typically thought to be much larger than the average. Mathematicians soon hypothesized that as your set gets bigger, the biggest sum-free subsets will get much larger than N/3.