Indestructible colourings and rainbow Ramsey theorems

英文摘要：

We give a negative answer to a question of Erdos and Hajnal: it is consistent that GCH holds and there is a colouring $c:[{ω_2}]^2\to 2$ establishing $ω_2 \not\to [(ω_1;ω)]^2_2$ such that some colouring $g:[ω_1]^2\to 2$ can not be embedded into $c$. It is also consistent that $2^{ω_1}$ is arbitrarily large, and a function $g$ establishes $2^{ω_1} \not\to [(ω_1,ω_2)]^2_{ω_1}$ such that there is no uncountable $g$-rainbow subset of $2^{ω_1}$. We also show that for each $k\in ω$ it is consistent with Martin's Axiom that the negative partition relation $ω_1 \not\to^* [(ω_1;ω_1)]_{k-bdd}$ holds.