Ramsey numbers for multiple copies of sparse graphs

For a graph H $H$ and an integer n $n$, we let n H $nH$ denote the disjoint union of n $n$ copies of H $H$. In 1975, Burr, Erdos and Spencer initiated the study of Ramsey numbers for n H $nH$, one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant c = c( H ) $c=c(H)$ such that r(n H ) =(2 divide H divide - alpha( H ) ) n + c $r(nH)=(2| H| -\alpha (H))n+c$, provided n $n$ is sufficiently large. Subsequently, Burr gave an implicit way of computing c $c$ and noted that this long-term behaviour occurs when n $n$ is triply exponential in divide H divide $| H| $. Very recently, Bucic and Sudakov revived the problem and established an essentially tight bound on n $n$ by showing r(n H ) $r(nH)$ follows this behaviour already when the number of copies is just a single exponential. We provide significantly stronger bounds on n $n$ in case H $H$ is a sparse graph, most notably of bounded maximum degree. These are relatable to the current state-of-the-art bounds on r( H ) $r(H)$ and (in a way) tight. Our methods rely on a beautiful classic proof of Graham, Rodl and Rucinski, with an emphasis on developing an efficient absorbing method for bounded degree graphs.