Abstract

It is consistent that there is a set mapping from the four-tuples of $\omega_n$ into the finite subsets with no free subsets of size t$_n$ for some natural number t$_n$. For any $n < \omega$ it is consistent that there is a set mapping from the pairs of $\omega_n$ into the finite subsets with no infinite free sets. For any $n < \omega$ it is consistent that there is a set mapping from the pairs of $\omega_n$ into $\omega_n$ with no uncountable free sets.