Asymmetric Ramsey numbers when the tree orders differ substantially

Characterize the Ramsey numbers R(T,S) and determine whether the lower bound \underline{R}(T,S) is exact for families of pairs of trees in which the order of T is significantly larger than the order of S.

Background

The positive results in the paper concern trees whose orders are comparable: the assumptions n\geq\nu and \nu\geq t_1 imply n\geq\nu\geq n/2. The authors explicitly raise the unresolved issue of what occurs when |T| is substantially larger than |S|. They note that bounded-degree trees versus general graphs provide one special positive result, but broader families remain to be identified.