Asymptotic equality between set-coloring Ramsey numbers and code sizes

Determine whether, for every fixed prime power q and constant C>0, the set-coloring Ramsey number R(q+1;r,s) satisfies R(q+1;r,s)=(1+o(1))A_q(r,s) uniformly over positive integers r and s with 0<=(1-1/q)r-s<=C r^{1/3} as r tends to infinity.

Background

The paper uses the general relationship between set-coloring Ramsey numbers and q-ary code sizes to obtain matching polynomial orders, but its Ramsey upper bound loses a constant factor through the distance-reduction step in the Conlon–Fox–Pham–Zhao conversion theorem. Thus, the results do not establish equality, even asymptotically, between the Ramsey number and the corresponding maximum code size.

A conjectural relation of the form R(q+1;r,s)=A_q(r,s)+1 had previously been suggested when s is sufficiently close to (1-1/q)r, without a precise parameter range. The paper formulates an explicit unresolved range in which the deficit from the zero-rate threshold is at most of order r^{1/3}. Along the paper's constructed sequence, a positive answer would imply the sharper asymptotic R(q+1;r,s)=(1+o(1))Q^4.