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Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then \[R(G,H)\ll m?\]
In other words, is $G$ Ramsey size linear? This fails for a graph $G$ with $n$ vertices and $2n-2$ edges (for example with $H=K_n$). Erdős, Faudree, Rousseau, and Schelp [EFRS93] have shown that any graph $G$ with $n$ vertices and at most $n+1$ edges is Ramsey size linear.

Implies [567].

See also the entry in the graphs problem collection.