We show that if a colouring c establishes ω2→[(ω1; ω)] 2 ω then c establishes this negative partition relation in each Cohen-generic extension of the ground model, i.e. this property of c is Cohen-indestructible. This result yields a negative answer to a question of Erdős and Hajnal: it is consistent that GCH holds and there is a colouring c: [ ] 2 ω2 − → 2 establishing ω2→[(ω1; ω)] 2 2 such that some colouring g: [ ] 2 ω1 − → 2 can not be embedded into c. It is also consistent that 2ω1 is arbitrarily large, and there is a function g establishing 2ω1→[(ω1, ω2)] 2 ω1 but there is no uncountable g-rainbow subset of 2ω1. We also show that if GCH holds then for each k ∈ ω there is a k-bounded colouring f: [ ] 2 ω1 → ω1 and there are two c.c.c posets P and Q such that but V P | = “f c.c.c-indestructibly establishes ω1 → ∗ [(ω1; ω1)]k−bdd”, V Q | = “ ω1 is the union of countably many f-rainbow sets ”
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