Partitions of large Rado graphs

Abstract

Let κ be a cardinal which is measurable after generically adding ${\beth_{\kappa+\omega}}$ many Cohen subsets to κ and let ${\mathcal G= ( \kappa,E )}$ be the κ-Rado graph. We prove, for 2 ≤ m < ω, that there is a finite value ${r_m^+}$ such that the set [κ]^m can be partitioned into classes ${\langle{C_i:i<r^{+} _m}\rangle}$ such that for any coloring of any of the classes C _i in fewer than κ colors, there is a copy ${\mathcal{G} ^\ast}$ of ${\mathcal G}$ in ${\mathcal G}$ such that ${[\mathcal{G} ^\ast ] ^m\cap C_i}$ is monochromatic. It follows that ${\mathcal{G}\rightarrow (\mathcal{G} ) ^m_{<\kappa/r_m^+}}$, that is, for any coloring of ${[ \mathcal {G} ] ^m}$ with fewer than κ colors there is a copy ${\mathcal{G} ^{\prime}}$ of ${\mathcal{G}}$ such that ${[\mathcal{G} ^{\prime} ] ^{m}}$ has at most ${r_m^+}$ colors. On the other hand, we show that there are colorings of ${\mathcal{G}}$ such that if ${\mathcal{G} ^{\prime}}$ is any copy of ${\mathcal{G}}$ then ${C_i\cap [\mathcal{G} ^{\prime} ] ^m\not=\emptyset}$ for all ${i<r^{+} _m}$, and hence ${\mathcal{G}\nrightarrow [ \mathcal{G} ] ^{m} _{r^{+} _m}}$ . We characterize ${r_m^+}$ as the cardinality of a certain finite set of types and obtain an upper and a lower bound on its value. In particular, ${r_2^+=2}$ and for m > 2 we have ${r_m^+ > r_m}$ where r _m is the corresponding number of types for the countable Rado graph.