Ramsey number of a cycle versus a graph of a given size

TL;DR

The paper addresses the asymmetric Ramsey problem for cycles versus graphs without isolated vertices by proving that for odd $k\ge 7$ and large $e(H)$, $R(C_k,H)\le 2e(H)+\left\lfloor\frac{k-1}{2}\right\rfloor$, settling a long-standing question of Erdős, Faudree, Rousseau and Schelp. The authors develop an induction on $e(H)$, leveraging strong bounds for $R(P_k,H)$ and a neighborhood-decomposition strategy around a vertex with large red degree to partition the host graph into two blue regions connected by blue edges. A key ingredient is a second-neighbourhood analysis: if the second neighbourhood contains a long path, a $C_k$ is forced, enabling the embedding framework. They also provide the exact matching-case bound and extend prior results for even $k$, contributing new techniques that connect path Ramsey bounds with structural neighbourhood arguments. The work advances asymmetric Ramsey theory for cycles and offers tools likely applicable to related cycle-vs-graph Ramsey problems with large edge counts.

Abstract

In this paper, we prove that for every $k$ and every graph $H$ with $m$ edges and no isolated vertices, the Ramsey number $R(C_k,H)$ is at most $2m+\lfloor \frac{k-1}{2} \rfloor$, provided $m$ is sufficiently large with respect to $k$. This settles a problem of Erdős, Faudree, Rousseau and Schelp.

Figures (2)

• Figure 1: From left to right, the cases $u_j\neq u_{j+k-4}$, $u_1\neq u_2$ and $u_1=u_{k-1}$ in the proof of \ref{['lemma:path_second_neighbourhood']}.
• Figure 2: In the proof of Theorem \ref{['thm:main+']} we fix a vertex $u$ in our host graph with a large red neighbourhood $U_1$. By Lemma \ref{['lemma:ramsey_of_path_vs_any']} we can embed a large part $H_1$ of $H$ into $U_1$. We can then show that either the second red neighbourhood $\Pi$ of $u$ has size at least $R(P_{2k}, H)$ --- in which case we can find a copy of $H$ in $\Pi$ --- or $U_2 = V(G) \setminus(U_1 \cup \Pi \cup \{u \})$ is large enough for us to find the rest of $H$ in there by induction. Note that by definition all the edges between $U_1$ and $U_2$ are blue.

Theorems & Definitions (19)

• Theorem 1: Goddard and Kleitman goddard1994upper; Sidorenko Sidorenko93
• Theorem 3
• Proposition 4: EFRS78Sudakov02
• Lemma 5: BEFRS82FL25
• Lemma 6
• proof
• Corollary 7
• Lemma 8
• proof
• Proposition 9: Matching case