Problem 568

Erdős Problem #568: declared status 'open'. Formalized: no. Let $G$ be a graph such that $R(G,T_n)\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\ll n^2$. Is it true that, for any $H$ with $m$ edges and no isolated vertices,$$R(G,H)\ll m?$$ Prize: no. Tags: graph theory, ramsey theory.