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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Coloring finite subsets of uncountable sets
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by Péter Komjáth and Saharon Shelah PDF
Proc. Amer. Math. Soc. 124 (1996), 3501-3505 Request permission

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
  • MR Author ID: 104465
  • Email: kope@cs.elte.hu
  • Saharon Shelah
  • Affiliation: Institute of Mathematics, Hebrew University, Givat Ram, 91904, Jerusalem, Israel
  • MR Author ID: 160185
  • ORCID: 0000-0003-0462-3152
  • Email: shelah@math.huji.ac.il
  • 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. 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
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 3501-3505
  • MSC (1991): Primary 03E35
  • DOI: https://doi.org/10.1090/S0002-9939-96-03450-8
  • MathSciNet review: 1342032