Ramsey numbers for sparse graphs Original Research Article

We consider a class of graphs on n vertices, called (d, f)-arrangeable graphs. This class of graphs contains all graphs of bounded degree d, and all df-arrangeable graphs, a class introduced by Chen and Schelp in 1993. In 1992, a variation of the Regularity Lemma of Szemerédi was introduced by Eaton and Rödl. As an application of this lemma, we give a linear upper bound, c(d, f)n, for the Ramsey number of graphs in this class, where log2 log2 c(d, f) = 24df5. This improves the earlier result, given in 1983 by Chvátal et al. of a linear bound on the Ramsey number of graphs with bounded degree d, where the constant term was more that a tower of d 2ʹs, and later extended by Chen and Schelp to include d-arrangeable graphs.