The tripartite Ramsey number for trees

We prove that for every ε>0 there are α>0 and n0∈N such that for all n⩾n0 the following holds. For any two-colouring of the edges of Kn,n,n one colour contains copies of all trees T of order k⩽(3−ε)n/2 and with maximum degree Δ(T)⩽nα. This answers a conjecture of Schelp.