### Coloring finite subsets of uncountable sets

by Komjath and Shelah. [KoSh:516]

Proc American Math Soc, 1996

It is consistent for every (1 <= n< omega) that
(2^omega = omega_n) and there is a function
(F:[omega_n]^{< omega}-> omega) such that every finite set
can be written 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
(sum^n_{i=1}{n+i choose n}{n choose i}) ways as the union of
two sets with the same color.

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