Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Partitions and sums and products of integers


Author: Neil Hindman
Journal: Trans. Amer. Math. Soc. 247 (1979), 227-245
MSC: Primary 10A45; Secondary 05A17, 54A25
DOI: https://doi.org/10.1090/S0002-9947-1979-0517693-4
MathSciNet review: 517693
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The principal result of the paper is that, if $ r\, < \,\omega $ and $ {\{ {A_i}\} _{i < r}}$ is a partition of $ \omega $, then there exist $ i\, < \,r$ and infinite subsets B and C of $ \omega $ such that $ \sum F\, \in \,{A_i}$ and $ \prod {G\, \in \,{A_i}} $ whenever F and G are finite nonempty subsets of B and C respectively. Conditions on the partition are obtained which are sufficient to guarantee that B and C can be chosen equal in the above statement, and some related finite questions are investigated.


References [Enhancements On Off] (What's this?)

  • [1] J. Baumgartner, A short proof of Hindman's theorem, J. Combinatorial Theory Ser. A 17 (1974), 384-386. MR 0354394 (50:6873)
  • [2] W. Comfort, Some recent applications of ultrafilters to topology, (Proc. Fourth 1976 Prague Topological Symposium), Lecture Notes in Math., Springer-Verlag, Berlin and New York, 1978. MR 0451187 (56:9474)
  • [3] -, Ultrafilters: Some old and some new results, Bull. Amer. Math. Soc. 83 (1977), 417-455. MR 0454893 (56:13136)
  • [4] L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton, N. J., 1960. MR 0116199 (22:6994)
  • [5] S. Glazer, Ultrafilters and semigroup combinatorics, J. Combinatorial Theory Ser. A (to appear).
  • [6] N. Hindman, Finite sums from sequences within cells of a partition of N, J. Combinatorial Theory Ser. A 17 (1974), 1-11. MR 0349574 (50:2067)
  • [7] -, Partitions and sums of integers with repetition, J. Combinatorial Theory Ser. A (to appear).
  • [8] -, The existence of certain ultrafilters on N and a conjecture of Graham and Rothschild, Proc. Amer. Math. Soc. 36 (1972), 341-346. MR 0307926 (46:7041)
  • [9] A. Tarski, Sur la décomposition des ensembles en sous-ensembles presque disjoints, Fund. Math. 12 (1928), 188-205.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 10A45, 05A17, 54A25

Retrieve articles in all journals with MSC: 10A45, 05A17, 54A25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0517693-4
Keywords: Partitions, ultrafilters, sums, products
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society