Refine the upper bound on n₀ for eventual strictness of the ξ_n–θ_n inequality

Improve the upper bound or determine the exact value of n₀, the minimal natural number such that for all n ≥ n₀ the strict inequality ξ_n(Q_n) < ((n+1)/2)(θ_n(Q_n) − 1) + 1 holds, where Q_n = [0,1]^n.

Background

Using asymptotic estimates ξ_n(Q_n) ≍ n and θ_n(Q_n) ≥ c√n, the authors show that the inequality becomes strict for sufficiently large n and define n₀ as the threshold dimension beyond which strictness persists.

Existing bounds are 8 ≤ n₀ ≤ 53, with the upper bound improved from earlier work (≤57) to ≤53. Tightening this upper bound or determining n₀ exactly remains unresolved.