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\[R(G,H)\ll m?\]
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
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].
This problem is
#31 in Ramsey Theory in the graphs problem collection.
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This page was last edited 18 January 2026. View history
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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 2026-04-11
