Eventual dominance of the Sphere method

Determine whether, for every integer n at least 10^9, the Sphere method produces a larger 3-free subset of [n] than each of the other methods considered in the paper.

Background

The paper compares several constructions of large 3-free subsets of [n], including the Base 3, Base 5, KD, Block, and Sphere methods. Empirical tables show that the Sphere method overtakes the other methods around n = 109 and is asymptotically superior according to the bounds surveyed.

The authors do not prove that this empirical ordering persists for every larger n. They formulate an explicit suspicion that the Sphere method remains the best construction beyond the observed computational range, making eventual dominance an unresolved comparative question.

References

Given this evidence and that asymptotically the Sphere method produces larger 3-free sets, we suspect that for n ≥ 109 the Sphere method really does produce larger 3-free sets of [n] than the other methods.

— Finding Large Sets Without Arithmetic Progressions of Length Three: An Empirical View and Survey II  (2501.01634 - Gasarch et al., 3 Jan 2025) in Section 5, Comparing All the Methods