Indestructible colourings and rainbow Ramsey theorems

Abstract

We give a negative answer to a question of Erdos and Hajnal: it is consistent that GCH holds and there is a colouring $c:[{\omega_2}]^2\to 2$ establishing $\omega_2 \not\to [(\omega_1;{\omega})]^2_2$ such that some colouring $g:[\omega_1]^2\to 2$ can not be embedded into $c$. It is also consistent that $2^{\omega_1}$ is arbitrarily large, and a function $g$ establishes $2^{\omega_1} \not\to [(\omega_1,\omega_2)]^2_{\omega_1}$ such that there is no uncountable $g$-rainbow subset of $2^{\omega_1}$. We also show that for each $k\in {\omega}$ it is consistent with Martin's Axiom that the negative partition relation $\omega_1 \not\to^* [(\omega_1;\omega_1)]_{k-bdd}$ holds.