Infinitary combinatorics: Difference between revisions
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combinatorics of the continuum and of successors of singular cardinals |
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In mathematics, '''infinitary combinatorics''', or '''combinatorial set theory''', is an extension of ideas in [[combinatorics]] to [[infinite set]]s. |
In mathematics, '''infinitary combinatorics''', or '''combinatorial set theory''', is an extension of ideas in [[combinatorics]] to [[infinite set]]s. |
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Some of the things studied include [[continuous graph]]s and [[tree (set theory)|trees]], extensions of [[Ramsey's theorem]], and [[Martin's axiom]]. |
Some of the things studied include [[continuous graph]]s and [[tree (set theory)|trees]], extensions of [[Ramsey's theorem]], and [[Martin's axiom]]. |
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Recent developments concern combinatorics of the continuum<ref> [[Andreas Blass]], ''Combinatorial Cardinal Characteristics of the Continuum'', Chapter 6 in Handbook of Set Theory, edited by [[Matthew Foreman]] and [[Akihiro Kanamori]], Springer, 2010</ref> and combinatorics on successors of singular cardinals <ref> Todd Eisworth, ''Successors of Singular Cardinals'' |
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Chapter 15 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010</ref>. |
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==Ramsey theory for infinite sets== |
==Ramsey theory for infinite sets== |
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|url=http://www.ams.org/bull/1956-62-05/S0002-9904-1956-10036-0/|doi=10.1090/S0002-9904-1956-10036-0}} |
|url=http://www.ams.org/bull/1956-62-05/S0002-9904-1956-10036-0/|doi=10.1090/S0002-9904-1956-10036-0}} |
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*{{Citation | last1=Kunen | first1=Kenneth | author1-link=Kenneth Kunen | title=[[Set Theory: An Introduction to Independence Proofs]] | publisher=North-Holland | location=Amsterdam | isbn=978-0-444-85401-8 | year=1980}} |
*{{Citation | last1=Kunen | first1=Kenneth | author1-link=Kenneth Kunen | title=[[Set Theory: An Introduction to Independence Proofs]] | publisher=North-Holland | location=Amsterdam | isbn=978-0-444-85401-8 | year=1980}} |
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==Notes== |
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<references> |
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[[Category:Set theory]] |
[[Category:Set theory]] |
Revision as of 21:47, 14 June 2011
In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum[1] and combinatorics on successors of singular cardinals [2].
Ramsey theory for infinite sets
Write κ, λ for ordinals, and m for a cardinal number and n for a natural number. Erdős & Rado (1956) introduced the notation
as a shorthand way of saying that every partition of the set [κ]n of n-element subsets of κ into m pieces has a homogeneous set of order type λ. A homogeneous set is one such that every n-element subset is in the same element of the partition. When m is 2 it is often omitted.
There are no ordinals κ with κ→(ω)ω, so n is usually taken to be finite. An extension where n is almost allowed to be infinite is the notation
which is a shorthand way of saying that every partition of the set of finite subsets of κ into m pieces has a subset of order type λ such that for any finite n, all subsets of size n are in the same element of the partition. When m is 2 it is often omitted.
Another variation is the notation
which is a shorthand way of saying that every coloring of the set [κ]n of n-element subsets of κ with 2 colors has a subset of order type λ such that all elements of [λ]n have the first color, or a subset of order type μ such that all elements of [μ]n have the second color.
Some properties of this include: (in what follows is a cardinal)
- for all finite n and k (Ramsey's theorem).
- (Erdős–Rado theorem.)
- (Erdős–Dushnik–Miller theorem).
Large cardinals
Several large cardinal properties can be defined using this notation. In particular:
- Weakly compact cardinals κ are those that satisfy κ→(κ)2
- α-Erdos cardinals κ are the smallest that satisfy κ→(α)<ω
- Ramsey cardinals κ are those that satisfy κ→(κ)<ω
References
- Dushnik, Ben; Miller, E. W. (1941), "Partially ordered sets", American Journal of Mathematics, 63 (3): 600–610, doi:10.2307/2371374, ISSN 0002-9327, MR0004862
- Erdős, Paul; Hajnal, András (1971), "Unsolved problems in set theory", Axiomatic Set Theory ( Univ. California, Los Angeles, Calif., 1967), Proc. Sympos. Pure Math, vol. XIII Part I, Providence, R.I.: Amer. Math. Soc., pp. 17–48, MR0280381
- Erdős, Paul; Hajnal, András; Máté, Attila; Rado, Richard (1984), Combinatorial set theory: partition relations for cardinals, Studies in Logic and the Foundations of Mathematics, vol. 106, Amsterdam: North-Holland Publishing Co., ISBN 0-444-86157-2, MR0795592
{{citation}}
: CS1 maint: extra punctuation (link) - Erdős, P.; Rado, R. (1956), "A partition calculus in set theory.", Bull. Amer. Math. Soc., 62: 427–489, doi:10.1090/S0002-9904-1956-10036-0, MR0081864
- Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, Amsterdam: North-Holland, ISBN 978-0-444-85401-8
Notes
<references>
- ^ Andreas Blass, Combinatorial Cardinal Characteristics of the Continuum, Chapter 6 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010
- ^ Todd Eisworth, Successors of Singular Cardinals Chapter 15 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010