Ramsey numbers of sparse graphs versus disjoint books

Let Bₖ denote a book on k+2 vertices and tBₖ be t vertex-disjoint Bₖ's. Let G be a connected graph with n vertices and at most n (1+ε) edges, where ε is a constant depending on k and t. In this paper, we show that the Ramsey number r (G, tBₖ) =2n+t-2 provided n 111t³k³. 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.