Indestructible colourings and rainbow Ramsey theorems

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Abstract: We give a negative answer to a question of Erdos and Hajnal: it is consistent that GCH holds and there is a colouring

c : [o m e g a_{2}]^{2} o 2

establishing

o m e g a_{2} o t o [(o m e g a_{1}; o m e g a)]_{2}^{2}

such that some colouring

g : [o m e g a_{1}]^{2} o 2

can not be embedded into

c

. It is also consistent that

2^{o m e g a_{1}}

is arbitrarily large, and a function

g

establishes

2^{o m e g a_{1}} o t o [(o m e g a_{1}, o m e g a_{2})]_{o m e g a_{1}}^{2}

such that there is no uncountable

g

-rainbow subset of

2^{o m e g a_{1}}

. We also show that for each

k i n o m e g a

it is consistent with Martin's Axiom that the negative partition relation

o m e g a_{1} o t o^{*} [(o m e g a_{1}; o m e g a_{1})]_{k - b d d}

holds.

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