Approximate tightness without bipartition-size assumptions

Determine whether the approximate Ramsey-number bound R(T,S)\leq(1+\mu)\underline{R}(T,S) holds, possibly under a stronger maximum-degree condition, when either the assumption s_2\geq t_2 or the assumption \nu\geq t_1 is omitted.

Background

The approximate theorem proves near-tightness of the lower bound \underline{R}(T,S) for trees with linear maximum degree, provided that s_2\geq t_2 and \nu\geq t_1. The paper constructs examples showing that dropping either assumption can increase the Ramsey number by an arbitrarily large additive amount. It remains unresolved whether a multiplicative approximate bound can nevertheless survive after removing one of these assumptions, perhaps with stronger degree restrictions.