The tripartite Ramsey number for trees

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