Online research in philosophy

We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal $\mu$ , of cofinality $\omega$ , such that every $\mu^{+}$ -chromatic graph X on $\mu^{+}$ has an edge colouring c of X into $\mu$ colours for which every vertex colouring g of X into at most $\mu$ many colours has a g-colour class on which c takes every value. The paper also contains some generalisations of the above statement in which $\mu^{+}$ is replaced (...) by other cardinals < $\mu$. (shrink)

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

It is consistent that there exists a set mapping $F: \lbrack\omega_2\rbrack^2 \rightarrow \lbrack\omega_2\rbrack^{<\omega}$ such that $F(\alpha, \beta) \subseteq \alpha$ for $\alpha < \beta < \omega_2$ and there is no infinite free subset for F. This solves a problem of A. Hajnal and A. Mate.