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 ${\omega }_2\nrightarrow [({\omega }_1;\omega ){]}_2^2$ 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 $2^{{\omega }_1}\nrightarrow [({\omega }_1,{\omega }_2){]}_{{\omega }_1}^2$ 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 ${\omega }_1/{\to }^{\ast }[({\omega }_1;{\omega }_1){]}_{k-bdd}$ holds.