Ramsey numbers of sparse graphs versus disjoint books

Let $B_k$ denote a book on $k+2$ vertices and $t{B}_k$ be $t$ vertex-disjoint $B_k$’s. Let $G$ be a connected graph with $n$ vertices and at most $n(1+\epsilon )$ edges, where $\epsilon$ is a constant depending on $k$ and $t$. In this paper, we show that the Ramsey number $$r(G,t{B}_k)=2n+t-2$$ provided $n\ge 111t^3{k}^3$. Our result extends the work of Erdős, Faudree, Rousseau, and Schelp (1988), who established the corresponding result for $G$ being a tree and $t=1$.