On a Conjecture of Rِdl and Voigt

We prove that if λ is an infinite cardinal number and G is any graph of cardinality κ = λ+ which is a union of a finite number of forests, then there is a graph Hk of size κ (which does not depend upon G) so that Hκ → (G)1λ. Rödl and Voight conjectured that there is such a graph Hκ for the special case when G is the regular tree on κ in which every vertex has degree κ. We also prove that if a graph is the union of n forests, then it has colouring number 2n.