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Abstract

We show that if a colouring $c$ establishes $\omega_2\nrightarrow [{(\omega_1:{\omega})}]^2$ 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 $c:[{\omega_2}]^2\to 2$ establishing $\omega_2\nrightarrow [{(\omega_1:{\omega})}]_2$ such that some colouring $g:[\omega_1]^2\to 2$ does not embed into $c$.

It is also consistent that $2^{\omega_1}$ is arbitrarily large, and there is a function $g$ establishing $2^{{\omega}_1}\nrightarrow [{(\omega_1,\omega_2)}]_{\omega_1};$ but there is no uncountable $g$-rainbow subset of $2^{{\omega}_1}$.

We also show that if GCH holds then for each $k\in {\omega}$ there is a $k$-bounded colouring $f:[\omega_1]^2\rightarrow \omega_1$ and there are two c.c.c. posets ${\cal P}$ and ${\cal Q}$ such that $$ V^{{\cal P}}\models \hbox{$f$ c.c.c.-indestructibly establishes $\omega_1\nrightarrow^* [(\omega_1;\omega_1)]_{k\hbox{-}{\rm bdd}}$,} $$ but $$ V^{{\cal Q}}\models \hbox{$\omega_1$ is the union of countably many $f$-rainbow sets.} $$