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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Coloring finite subsets of uncountable sets

Authors: Péter Komjáth and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 124 (1996), 3501-3505
MSC (1991): Primary 03E35
MathSciNet review: 1342032
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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.

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Additional Information

Péter Komjáth
Affiliation: Department of Computer Science, Eötvös University, Budapest, Múzeum krt. 6–8, 1088, Hungary

Saharon Shelah
Affiliation: Institute of Mathematics, Hebrew University, Givat Ram, 91904, Jerusalem, Israel

Keywords: Axiomatic set theory, independence proofs, combinatorial set theory
Received by editor(s): March 31, 1995
Received by editor(s) in revised form: May 9, 1995
Additional Notes: The first author was supported by the Hungarian OTKA, Grant No. T014105.\endgraf This paper is number 516 in the cumulative list of Shelah’s papers. His research was supported by the Basic Research Foundation of the Israel Academy of Sciences and Humanities.
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1996 American Mathematical Society