Ramsey numbers for sparse graphs
Date of Original Version
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 2's, and later extended by Chen and Schelp to include d-arrangeable graphs. © 1998 Elsevier Science B.V. All rights reserved.
Eaton, Nancy. "Ramsey numbers for sparse graphs." Discrete Mathematics 185, 1-3 (1998): 63-75. doi:10.1016/S0012-365X(97)00184-2.