The size-Ramsey number of trees

Given a graph G, the size-Ramsey number r̂(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 β(·) for trees and showed that r̂(T ) = Ω(β(T )). Moreover he conjectured that r̂(T ) = Θ(β(T )). We settle this conjecture by providing a family of graphs and an embedding scheme for trees.