Coloring finite subsets of uncountable sets

Abstract:

It is consistent for every $1\leq n< \omega$ that $2^{\omega }=\omega _{n}$ and there is a function $F:[\omega _{n}]^{< \omega }\to \omega$ such that every finite set can be written in at most $2^{n}-1$ ways as the union of two distinct monocolored sets. If GCH holds, for every such coloring there is a finite set that can be written at least $\frac {1}{2}\sum ^{n}_{i=1}{\binom {n+i}{n}}{\binom {n}{i}}$ ways as the union of two sets with the same color.