The size-Ramsey number of trees

IfG andH are graphs, let us writeG→(H)₂ ifG contains a monochromatic copy ofH in any 2-colouring of the edges ofG. Thesize-Ramsey number r _e(H) of a graphH is the smallest possible number of edges a graphG may have ifG→(H)₂. SupposeT is a tree of order |T|≥2, and lett ₀,t ₁ be the cardinalities of the vertex classes ofT as a bipartite graph, and let Δ(T) be the maximal degree ofT. Moreover, let Δ₀, Δ₁ be the maxima of the degrees of the vertices in the respective vertex classes, and letβ(T)=T ₀Δ₀+t ₁Δ₁. Beck 7] proved thatβ(T)/4≤r _e(T)=O{β(T)(log|T|)¹²}, improving on a previous result of his 6] stating thatr _e(T)≤Δ(T)|T|(log|T|)¹². In 6], Beck conjectures thatr _e(T)=O{Δ(T)|T|}, and in 7] he puts forward the stronger conjecture thatr _e(T)=O{β(T)}. Here, we prove the first of these conjectures, and come quite close to proving the second by showing thatr _e(T)=O{β(T)logΔ(T)}.