Ramsey numbers for sparse graphs versus path or cycle

For any integer $k\ge3$, let $G$ be a connected graph with $n\ge Ω(k^4)$ vertices and no more than $(1+\frac{1}{ck^2})n$ edges where $c>0$ is a constant, and $P_k$ or $C_k$ a path or cycle with $k$ vertices. In this paper we prove that if $k$ is odd then $r(G,C_k)=2n-1$. Moreover, \[ r(G,P_k)=\max\left\{n+\left\lfloor\frac k2\right\rfloor-1, n+k-2-α'-γ\right\}, \] where $α'$ is the independence number of an appropriate subgraph of $G$ and $γ=0$ if $k-1$ divides $n+k-3-α'$ and $γ=1$ otherwise. Our bound on $n$ in terms of $k$ significantly improves upon the previous bounds $n\ge Ω(k^{10})$ (from the first result) and $n\ge Ω(k^{12})$ (from the second result) established by Burr, Erdős, Faudree, Rousseau and Schelp ({\em Trans. Amer. Math. Soc.} 269 (1982), 501--512). % Moreover, the restriction on the number of edges in $G$ is relaxed. This is the first improvement in over 40 years.