Eventual dominance of the Block method over the KD method

Determine whether, for every integer n at least 10^90, the Block method produces a larger 3-free subset of [n] than the KD method.

Background

The Block and KD methods are two constructive approaches for generating large 3-free subsets of [n]. Although the Block method has better asymptotic bounds, the empirical comparison shows repeated crossovers because the Block method is particularly sensitive to the representation of n.

The data suggest that the Block method becomes permanently superior to the KD method near n = 1090, but the authors do not establish this transition mathematically. They explicitly state the corresponding suspicion as an unresolved question.

References

Given this evidence and that asymptotically BL is better than KD, we suspect that for n ≥ 1090 BL really does produces larger sets than KD.

— 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