Sh:516

• Komjáth, P., & Shelah, S. (1996). Coloring finite subsets of uncountable sets. Proc. Amer. Math. Soc., 124(11), 3501–3505. arXiv: math/9505216 DOI: 10.1090/S0002-9939-96-03450-8 MR: 1342032
• 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 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.
• Version 1995-05-12_10 (6p) published version (5p)

@article{Sh:516,
 author = {Komj{\'a}th, P{\'e}ter and Shelah, Saharon},
 title = {{Coloring finite subsets of uncountable sets}},
 journal = {Proc. Amer. Math. Soc.},
 fjournal = {Proceedings of the American Mathematical Society},
 volume = {124},
 number = {11},
 year = {1996},
 pages = {3501--3505},
 issn = {0002-9939},
 mrnumber = {1342032},
 mrclass = {03E05 (03E35)},
 doi = {10.1090/S0002-9939-96-03450-8},
 note = {\href{https://arxiv.org/abs/math/9505216}{arXiv: math/9505216}},
 arxiv_number = {math/9505216}
}