On strongly summable ultrafilters and union ultrafilters

Blass, Andreas; Hindman, Neil

doi:10.1090/S0002-9947-1987-0906807-4

On strongly summable ultrafilters and union ultrafilters
HTML articles powered by AMS MathViewer

by Andreas Blass and Neil Hindman PDF

Trans. Amer. Math. Soc. 304 (1987), 83-97 Request permission

Abstract:

We prove that union ultrafilters are essentially the same as strongly summable ultrafilters but ordered-union ultrafilters are not. We also prove that the existence of ultrafilters of these sorts implies the existence of $P$-points and therefore cannot be established in ZFC.

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
Murray G. Bell, On the combinatorial principle $P({\mathfrak {c}})$, Fund. Math. 114 (1981), no. 2, 149–157. MR 643555, DOI 10.4064/fm-114-2-149-157
Andreas Blass, Ultrafilters related to Hindman’s finite-unions theorem and its extensions, Logic and combinatorics (Arcata, Calif., 1985) Contemp. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1987, pp. 89–124. MR 891244, DOI 10.1090/conm/065/891244
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
R. L. Graham and B. L. Rothschild, Ramsey’s theorem for $n$-parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257–292. MR 284352, DOI 10.1090/S0002-9947-1971-0284352-8
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
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
Neil Hindman, Ultrafilters and combinatorial number theory, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 119–184. MR 564927
Neil Hindman, Summable ultrafilters and finite sums, Logic and combinatorics (Arcata, Calif., 1985) Contemp. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1987, pp. 263–274. MR 891252, DOI 10.1090/conm/065/891252
Katsumi Numakura, On bicompact semigroups, Math. J. Okayama Univ. 1 (1952), 99–108. MR 48467
Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955
Edward L. Wimmers, The Shelah $P$-point independence theorem, Israel J. Math. 43 (1982), no. 1, 28–48. MR 728877, DOI 10.1007/BF02761683