All Random Solved Random Open
For any $d\geq 1$ if $H$ is a graph such that every subgraph contains a vertex of degree at most $d$ then $R(H)\ll_d n$.
The Burr-Erdős conjecture. This is equivalent to showing that if $H$ is the union of $c$ forests then $R(H)\ll_c n$, and also that if every subgraph has average degree at most $d$ then $R(H)\ll_d n$. Solved by Lee [Le16], who proved that \[ R(H) \leq 2^{2^{O(d)}}n.\]

This problem is #9 in Ramsey Theory in the graphs problem collection. See also [800].