The Ramsey number of a graph with bounded maximum degree

Abstract The Ramsey number of a graph G is the least number t for which it is true that whenever the edges of the complete graph on t vertices are colored in an arbitrary fashion using two colors, say red and blue, then it is always the case that either the red subgraph contains G or the blue subgraph contains G . A conjecture of P. Erdos and S. Burr is settled in the affirmative by proving that for each d ≥ 1, there exists a constant c so that if G is any graph on n vertices with maximum degree d , then the Ramsey number of G is at most cn .