An exact Ramsey number of large bipartite graphs versus odd wheel

TL;DR

The paper resolves the exact Ramsey number R( K_{2,n}, W_m) for large bipartite graphs versus odd wheels, proving $R(\, K_{2,n}, W_m) = 3n+4$ when $n$ and $m$ are sufficiently large with $n  geq 4m$ and $m$ odd. The authors blend a probabilistic first-moment approach with detailed structural analysis of neighbor sets, focusing on the complement's neighborhood around a maximum-degree vertex to force a cycle $C_m$, and thus a wheel $W_m$. Key tools include the cycle-embedding lemmas of Haxell et al. and Allen et al., plus an intersection lemma guaranteeing large common neighborhoods, enabling a contradiction if a $ K_{2,n}$-free graph on $3n+4$ vertices were to exist. This result establishes the $W_m$-goodness of $ K_{2,n}$ and advances the understanding of Ramsey goodness for bipartite graphs against odd wheels, with potential applicability to broader $(G,H)$ pairs in Ramsey theory.

Abstract

The Ramsey number for the pair of graphs $\mathbb{K}_{1,n}$ (star) versus $W_{m}$ (wheel) has been extensively studied. In contrast, the Ramsey number of $\mathbb{K}_{2,n}$ versus wheel is not yet explored due to the structural complexity of $\mathbb{K}_{2,n}$. In this article, we have established an exact value of $\mathbb{K}_{2,n}$ versus $W_{m}$ for large $n$ and $m$. In particular, we have proved \begin{equation*} R(\mathbb{K}_{2,n}, W_{m})=3n+4 \end{equation*} if $n$ and $m$ are sufficiently large integers where $n\geq4m$ and $m$ is an odd integer. This also proves the $W_{m}$-goodness of $\mathbb{K}_{2,n}$. We have used a probabilistic method composed of structural analysis in our proof.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4