Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-04T23:00:26.050Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE EXISTENCE OF LARGE ANTICHAINS FOR DEFINABLE QUASI-ORDERS

Published online by Cambridge University Press:  10 December 2019

BENJAMIN D. MILLER  and
ZOLTÁN VIDNYÁNSZKY
BENJAMIN D. MILLER
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITÄT WIEN WÄHRINGER STRASSE 25 1090 WIEN AUSTRIAE-mail:benjamin.miller@univie.ac.atURL:http://www.logic.univie.ac.at/~millerb45/
ZOLTÁN VIDNYÁNSZKY
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITÄT WIEN WÄHRINGER STRASSE 25 1090 WIEN AUSTRIAE-mail:zoltan.vidnyanszky@univie.ac.atURL:http://www.logic.univie.ac.at/zoltan.vidnyanszky/
Get access Rights & Permissions [Opens in a new window]

Abstract

We simultaneously generalize Silver’s perfect set theorem for co-analytic equivalence relations and Harrington-Marker-Shelah’s Dilworth-style perfect set theorem for Borel quasi-orders, establish the analogous theorem at the next definable cardinal, and give further generalizations under weaker definability conditions.

Keywords

Primary 03E1528A05antichainchaindichotomyDilworthHarrington-Marker-Shelahquasi-ordersmooth
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 103 - 108
Copyright
Copyright © The Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Harrington, L. A., Kechris, A. S., and Louveau, A., A Glimm-Effros dichotomy for Borel equivalence relations. Journal of the American Mathematical Society, vol. 3 (1990), no. 4, pp. 903–928.CrossRefGoogle Scholar
Harrington, L. A., Marker, D. E., and Shelah, S., Borel orderings. Transactions of the American Mathematical Society, vol. 310 (1988), no. 1, pp. 293–302.CrossRefGoogle Scholar
Harrington, L. A. and Sami, R. L., Equivalence relations, projective and beyond, Logic Colloquium ’78 (Mons, 1978) (Boffa, M., van Dalen, D., and McAloon, K., editors), Studies in Logic and the Foundations of Mathematics, vol. 97, North-Holland, Amsterdam-New York, 1979, pp. 247–264.Google Scholar
Harrington, L. A. and Shelah, S., Counting equivalence classes for co-κ-Souslin equivalence relations, Logic Colloquium ’80 (Prague, 1980) (van Dalen, D., Lascar, D., and Smiley, T. J., editors), Studies in Logic and the Foundations of Mathematics, vol. 108, North-Holland, Amsterdam-New York, 1982, pp. 147–152.CrossRefGoogle Scholar
Jackson, S. C., Suslin cardinals, partition properties, homogeneity. Introduction to Part II, Games, Scales, and Suslin Cardinals. The Cabal Seminar. Vol. I (Kechris, A. S., Löwe, B., and Steel, J., editors), Lecture Notes in Logic, vol. 31, Association for Symbolic Logic, Chicago, IL, 2008, pp. 273–313.CrossRefGoogle Scholar
Kanovei, V. G., Two dichotomy theorems on colourability of non-analytic graphs. Fundamenta Mathematicae, vol. 154 (1997), no. 2, pp. 183–201. European Summer Meeting of the Association for Symbolic Logic (Haifa, 1995).Google Scholar
Kechris, A. S., On projective ordinals, this Journal, vol. 39 (1974), pp. 269–282.Google Scholar
Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
Kechris, A. S., Solecki, S. J., and Todorcevic, S., Borel chromatic numbers. Advances in Mathematics, vol. 141 (1999), no. 1, pp. 1–44.CrossRefGoogle Scholar
Martin, D. A., Pleasant and unpleasant consequences of determinateness, unpublished manuscript, March 1970.Google Scholar
Miller, B. D., The graph-theoretic approach to descriptive set theory. The Bulletin of Symbolic Logic, vol. 18 (2012), no. 4, pp. 554–575.CrossRefGoogle Scholar
Moschovakis, Y. N., Uniformization in a playful universe. Bulletin of the American Mathematical Society, vol. 77 (1971), pp. 731–736.CrossRefGoogle Scholar
Silver, J. H., Counting the number of equivalence classes of Borel and coanalytic equivalence relations. Annals of Mathematical Logic, vol. 18 (1980), no. 1, pp. 1–28.CrossRefGoogle Scholar