PROVED This has been solved in the affirmative. - $100

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\[\hat{R}(C_n) =o(n^2)?\]

#720: [Er76c,p.5][EFRS78b][Er81][Er82e]

graph theory | ramsey theory