Problem Statement
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\[R(G,H)\ll m?\]
Categories:
Graph Theory Ramsey Theory
Progress
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.
Source: erdosproblems.com/566 | Last verified: January 15, 2026