Indestructible colourings and rainbow Ramsey theorems

Lajos Soukup¹

¹ Alfréd Rényi Institute of Mathematics V. Reáltanoda utca 13-15 H-1053 Budapest, Hungary

Fundamenta Mathematicae, Tome 202 (2009) no. 2, pp. 161-180

Résumé

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 $$ V^{{\cal Q}}\models \hbox{$\omega_1$ is the union of countably many $f$-rainbow sets.} $$

Détail

DOI : 10.4064/fm202-2-4

Keywords: colouring establishes omega nrightarrow omega omega establishes negative partition relation each cohen generic extension ground model property cohen indestructible result yields negative answer question erd hajnal consistent gch holds there colouring omega establishing omega nrightarrow omega omega colouring omega does embed consistent omega arbitrarily large there function establishing omega nrightarrow omega omega omega there uncountable g rainbow subset omega gch holds each omega there k bounded colouring omega rightarrow omega there posets cal cal cal models hbox c indestructibly establishes omega nrightarrow * omega omega hbox bdd cal models hbox omega union countably many f rainbow sets