Problem Statement
Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$.
Is it true that, if $P_n$ is the path of length $n$, then\[\hat{R}(P_n)/n\to \infty\]and\[\hat{R}(P_n)/n^2 \to 0?\]Is it true that, if $C_n$ is the cycle with $n$ edges, then\[\hat{R}(C_n) =o(n^2)?\]
Is it true that, if $P_n$ is the path of length $n$, then\[\hat{R}(P_n)/n\to \infty\]and\[\hat{R}(P_n)/n^2 \to 0?\]Is it true that, if $C_n$ is the cycle with $n$ edges, then\[\hat{R}(C_n) =o(n^2)?\]
Categories:
Graph Theory Ramsey Theory
Progress
A problem of Erdős, Faudree, Rousseau, and Schelp [EFRS78b].Answered by Beck [Be83b], who proved that in fact $\hat{R}(P_n)\ll n$ and $\hat{R}(C_n)\ll n$.
A more general problem concerning the size Ramsey number of graphs with bounded maximum degree is [559].
Source: erdosproblems.com/720 | Last verified: January 16, 2026