The size Ramsey number of trees with bounded degree

Abstract

AbstractThe size Ramsey number r̂(G, H) of graphs G and H is the smallest integer r̂ such that there is a graph F with r̂ edges and if the edge set of F is red‐blue colored, there exists either a red copy of G or a blue copy of H in F. This article shows that r̂(Tnd, Tnd) ⩽ c · d2 · n and c · n3 ⩽ r̂(Kn, Tnd) ⩽ c(d)·n3 log n for every tree Tnd on n vertices. and maximal degree at most d and a complete graph Kn on n vertices. A generalization will be given. Probabilistic method is used throught this paper. © 1993 John Wiley Sons, Inc.