Maximum Subgraphs in Ramsey Graphs

Abstract

For graphs G, F and H, let \(G\rightarrow (F,H)\) signify that any edge coloring of G by red and blue contains either a red F or a blue H. Thus the Ramsey number R(F, H) is \(\min \{r\;|\; K_r\rightarrow (F,H)\}\). In this note, we consider an optimization problem as follows. For an integer \(k\ge 1\), let \({\mathbb {G}}=\{G_k,G_{k+1},\dots \}\) be a class of graphs \(G_n\) with \(\delta (G_n)\ge 1\). We define the critical Ramsey number \(R_{\mathbb {G}}(F,H)\) as \(\max \{n\;|\;K_r\setminus G_n \rightarrow (F,H),\;G_n\in \mathbb {G}\}\), where \(r=R(F,H)\). For some pairs F and H, we shall determine \(R_{\mathbb {G}}(F,H)\), where \({\mathbb {G}}\) consists of books, matchings and complete graphs, respectively.