Uniform polynomially superlinear growth across all lengths

Determine whether, for each fixed prime power q, there exists a constant C_q>0 such that, for every fixed B>C_q, the estimate R(q+1;r,(1-1/q)(r-j))=Theta_{q,B}(r^{4/3}) holds uniformly over positive integers r and j satisfying C_q r^{1/3}<=j<=B r^{1/3} and (1-1/q)(r-j) in the positive integers, as r tends to infinity.

Background

The paper establishes R(q+1;r,(1-1/q)(r-j))=Theta_q(r^{4/3}) for explicit sequences of lengths and deficits, and extends this order estimate to local ranges around those lengths. However, for fixed q, the constructed lengths have successive ratios tending to q^3, while the local ranges cover only deviations of order O(r_m^{1/3}). Consequently, the results do not address every sufficiently large length.

The paper notes that the upper bound in the proposed statement follows from the Balla-type coding estimate together with the Conlon–Fox–Pham–Zhao conversion inequality. The unresolved issue is constructing sufficiently large codes, and hence Ramsey lower bounds, for every sufficiently large length while retaining a deficit of order r^{1/3}.