Asymptotically optimal Ramsey goodness of sparse graphs versus odd cycles and paths
Chunchao Fan, Qizhong Lin
TL;DR
The paper addresses the sparse-graph Ramsey problem for odd cycles $C_k$ and paths $P_k$, aiming to minimize the vertex threshold $n$ relative to $k$. It introduces two new tools—the end-edge matching reconstruction lemma and an enhanced dichotomy lemma—and uses them to derive asymptotically tight bounds, first for odd cycles and then for paths, under near-linear edge-density. It establishes $r(G,C_k)=2n-1$ for suitable sparse $G$ with $n=Ω(k)$ and $e(G)≤(1+O(1/k^2))n$, and provides a general formula $r(G,P_k)= ext{max}ig\{ n+ig floor k/2ig floor-1, n+k-2- ext{α′}- ext{γ}igigig rac{ }{}}$ under two regimes, including $n=Ω(k^2)$ with $e(G)≤(1+O(1/k^2))n$ and $n=Ω(k)$ with additional degree/independence hypotheses. These results unify and extend classical theorems on cycles and paths, drastically improving prior polynomial gaps and clarifying the asymptotic Ramsey-goodness landscape for sparse graphs.
Abstract
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, Erdős, Faudree, Rousseau, and Schelp (Trans. AMS 1982) addressed this problem, establishing bounds requiring $n = Ω(k^{10})$ for odd cycles and $n = Ω(k^{12})$ for paths. We settle the asymptotic version of this problem, proving that these bounds are essentially tight: $n = Ω(k)$ suffices for odd cycles and $n = Ω(k^2)$ (or $n = Ω(k)$ 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 $n = Ω(k)$ and $e(G) \le (1 + O(1/k^2)) n$. (2) For paths $P_k$ ($k\ge2$), we prove $r(G, P_k) = \max\{ n + \lfloor k/2\rfloor - 1, n + k - 2 - α' - γ\}$ for any connected $n$-vertex graph $G$ satisfying one of the following: (i) $n = Ω(k^2)$ and $e(G) \le (1 + O(1/k^2)) n$; (ii) $n = Ω(k)$, $δ(G)\ge2$, $α'\geq k/2$, and $e(G) \le (1 + O(1/k)) n$. In the above, $α'$ 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. Consequently, our results unify and generalize classical theorems on odd cycles due to Bondy and Erdős (1973), Faudree and Schelp (1974), and Rosta (1973), and on paths due to Gerencsér and Gyárfás (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.