OPEN
This is open, and cannot be resolved with a finite computation.
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.
View the LaTeX source
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #566, https://www.erdosproblems.com/566, accessed 2026-01-16
