Eventual stabilization of the ranking of all construction methods

Determine whether, for every integer n at least 10^90, the asymptotic behavior of the Base 3, Base 5, KD, Block, and Sphere methods agrees with their empirical ranking in determining which method produces the largest, second-largest, and subsequent largest 3-free subsets of [n].

Background

The paper presents empirical rankings of five methods for constructing large 3-free subsets: Base 3, Base 5, KD, Block, and Sphere. The observed ordering changes at several values of n, reflecting finite-scale effects and sensitivity to the numerical representation of n.

Beyond approximately n = 1090, the tables appear to exhibit a stable ordering, and the asymptotic estimates suggest that this ordering may persist. The authors explicitly leave unresolved whether the asymptotic behavior will permanently match the empirical ranking for all methods.

References

More generally, we suspect that for n ≥ 1090 the asymptotic behavior will match the empirical behavior for all the methods with regard to which one produces the largest, second largest, etc 3-free sets.

— 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