Asymptotically optimal Ramsey goodness of sparse graphs versus odd cycles and paths

A fundamental problem in graph Ramsey theory is to determine, for sparse graphs $G$ on $n$ vertices, the minimal $n$ such that $G$ is Ramsey-good for odd cycles $C_k$ and paths $P_k$. Burr, Erd\H{o}s, Faudree, Rousseau, and Schelp (Trans. AMS 1982) addressed this problem, establishing bounds requiring $n = \Omega(k^{10})$ for odd cycles and $n = \Omega(k^{12})$ for paths. We settle the asymptotic version of this problem, proving that these bounds are essentially tight: $n = \Omega(k)$ suffices for odd cycles and $n = \Omega(k^2)$ (or $n = \Omega(k)$ under additional conditions) for paths. Specifically, we prove: (1) For odd cycles $C_k$ ($k\ge3$), we prove $r(G, C_k) = 2n-1$ for any connected $n$-vertex graph $G$ satisfying the relaxed conditions $n = \Omega(k)$ and $e(G) \le (1 + O(1/k^2)) n$. (2) For paths $P_k$ ($k\ge2$), we prove $r(G, P_k) = \max\{ n + \lfloor k/2\rfloor - 1, n + k - 2 - \alpha' - \gamma \}$ for any connected $n$-vertex graph $G$ satisfying one of the following: (i) $n = \Omega(k^2)$ and $e(G) \le (1 + O(1/k^2)) n$; (ii) $n = \Omega(k)$, $\delta(G)\ge2$, $\alpha'\geq k/2$, and $e(G) \le (1 + O(1/k)) n$. In the above, $\alpha'$ is the independence number of an appropriate subgraph of $G$ and $\gamma=0$ if $k-1$ divides $n+k-3-\alpha'$, and $\gamma=1$ otherwise. Consequently, our results unify and generalize classical theorems on odd cycles due to Bondy and Erd\H{o}s (1973), Faudree and Schelp (1974), and Rosta (1973), and on paths due to Gerencs\'er and Gy\'arf\'as (1967), Faudree, Lawrence, Parsons and Schelp (1974), and Parsons (1974). The proofs feature two key innovations: a novel reconstruction of the end-edge matching and an enhancement of Burr et al.'s dichotomy lemma.