A note on set mappings: an extension of results of Hajnal and Máté under GCH
Volume 246 / 2019
Abstract
A set mapping $f:[\lambda ]^r\to {\cal P}(\lambda )$ is of order $(\mu _0,\mu _1,\dots ,\mu _r)$ if for $s\in [\lambda ]^r$ increasingly enumerated as $\{\alpha _0,\alpha _1,\dots ,\alpha _{r-1}\}$ we have $|f(s)\cap (\alpha _i,\alpha _{i+1})| \lt \mu _{i+1}$ for $-1\leq i\leq r$ where $\alpha _{-1}=0$ and $\alpha _r=\lambda $. If GCH holds, $1\leq r \lt \omega $, $\kappa $ is an infinite cardinal, and $f:[\kappa ^{+r}]^r\to {\cal P}(\kappa ^{+r})$ is a set mapping of order $(\kappa ,\kappa ^+,\dots ,\kappa ^{+r})$, then there is a free set of order type $\kappa ^++r-1$. This is sharp in the sense that no free set of larger type can be guaranteed, and if any of the $\mu _i$’s is increased then even a free set of cardinality $\aleph _\omega $ cannot be guaranteed (for any $\kappa $).
