Partitions and sums and products of integers

Hindman, Neil

doi:10.1090/S0002-9947-1979-0517693-4

Partitions and sums and products of integers
HTML articles powered by AMS MathViewer

by Neil Hindman PDF

Trans. Amer. Math. Soc. 247 (1979), 227-245 Request permission

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

James E. Baumgartner, A short proof of Hindman’s theorem, J. Combinatorial Theory Ser. A 17 (1974), 384–386. MR 354394, DOI 10.1016/0097-3165(74)90103-4
W. W. Comfort, Some recent applications of ultrafilters to topology, General topology and its relations to modern analysis and algebra, IV (Proc. Fourth Prague Topological Sympos., Prague, 1976) Lecture Notes in Math., Vol. 609, Springer, Berlin, 1977, pp. 34–42. MR 0451187
W. W. Comfort, Ultrafilters: some old and some new results, Bull. Amer. Math. Soc. 83 (1977), no. 4, 417–455. MR 454893, DOI 10.1090/S0002-9904-1977-14316-4
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2

Ultrafilters and semigroup combinatorics

Neil Hindman, Finite sums from sequences within cells of a partition of $N$, J. Combinatorial Theory Ser. A 17 (1974), 1–11. MR 349574, DOI 10.1016/0097-3165(74)90023-5

Partitions and sums of integers with repetition

Neil Hindman, The existence of certain ultra-filters on $N$ and a conjecture of Graham and Rothschild, Proc. Amer. Math. Soc. 36 (1972), 341–346. MR 307926, DOI 10.1090/S0002-9939-1972-0307926-0

Sur la décomposition des ensembles en sous-ensembles presque disjoints

12