Coloring finite subsets of uncountable sets
Logic
2016-09-06 v1
Abstract
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.
Cite
@article{arxiv.math/9505216,
title = {Coloring finite subsets of uncountable sets},
author = {Peter Komjath and Saharon Shelah},
journal= {arXiv preprint arXiv:math/9505216},
year = {2016}
}