The size-Ramsey number of trees

Given a graph G, the size-Ramsey number (r) over cap (G) is the minimum number m for which there exists a graph F on m edges such that any two-coloring of the edges of F admits a monochromatic copy of G. In 1983, J. Beck introduced an invariant beta(.) for trees and showed that (r) over cap (T) = Omega(beta(T)). Moreover he conjectured that (r) over cap (T) = Theta(beta(T)). We settle this conjecture by providing a family of graphs and an embedding scheme for trees. (C) 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 49-73, 2012