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In mathematics, '''infinitary combinatorics''', or '''combinatorial set theory''', is an extension of ideas in [[combinatorics]] to [[infinite set]]s.
Some of the things studied include [[continuous graph]]s and [[tree (set theory)|trees]], extensions of [[Ramsey's theorem]], and [[Martin's axiom]].
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'' Chapter 15 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010</ref>


==Ramsey theory for infinite sets==
Write κ, λ for ordinals,  ''m'' for a [[cardinal number]] and ''n'' for a natural number. {{harvtxt|Erdős|Rado|1956}} introduced the notation
:<math>\kappa\rightarrow(\lambda)^n_m</math>
as a shorthand way of saying that every [[partition of a set|partition]] of the set [κ]<sup>''n''</sup> of ''n''-element subsets of <math>\kappa</math> into ''m'' pieces has a [[homogeneous set]] of order type λ. A homogeneous set is in this case a subset of κ such that every ''n''-element subset is in the same element of the partition. When ''m'' is 2 it is often omitted.


Assuming the [[Axiom of Choice]], there are no ordinals κ with κ→(ω)<sup>ω</sup>, so ''n'' is usually taken to be finite. An extension where ''n'' is almost allowed to be infinite is
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the notation
:<math>\kappa\rightarrow(\lambda)^{<\omega}_m</math>
which is a shorthand way of saying that every [[partition of a set|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
:<math>\kappa\rightarrow(\lambda, \mu)^n</math>
which is a shorthand way of saying that every coloring of the set [κ]<sup>''n''</sup> of ''n''-element subsets of κ with 2 colors has a subset  of order type λ such that all elements of [λ]<sup>''n''</sup> have the first color, or a subset  of order type μ such that all elements of [μ]<sup>''n''</sup> have the second color.  
 
Some properties of this include: (in what follows <math>\kappa</math> is a cardinal)
:<math>\alef_0\rightarrow(\alef_0)^n_k</math> for all finite ''n'' and ''k'' ([[Ramsey's theorem]]).
:<math>\beth_n^+\rightarrow(\alef_1)_{\alef_0}^{n+1}</math> ([[Erdős–Rado theorem]].)
:<math>2^\kappa\not\rightarrow(\kappa^+)^2</math> (Sierpiński theorem)
:<math>2^\kappa\not\rightarrow(3)^2_\kappa</math>
 
:<math>\kappa\rightarrow(\kappa,\alef_0)^2</math> (Erdős–Dushnik–Miller theorem).
 
In choiceless universes, partition properties with infinite exponents may hold, and some of them are obtained as consequences of the [[Axiom of determinacy]] (AD). For example, [[Donald A. Martin]] proved that AD implies 
:<math>\alef_1\rightarrow(\alef_1)^{\alef_1}_2</math>
 
==Large cardinals==
Several [[large cardinal]] properties can be defined using this notation. In particular:
*[[Weakly compact cardinal]]s κ are those that satisfy κ→(κ)<sup>2</sup>
*α-[[Erdős cardinal]]s κ are the smallest that satisfy κ→(α)<sup><ω</sup>
*[[Ramsey cardinal]]s κ are those that satisfy κ→(κ)<sup><ω</sup>
 
==References==
*{{Citation | last1=Dushnik | first1=Ben | last2=Miller | first2=E. W. | title=Partially ordered sets | jstor=2371374 | mr=0004862 | year=1941 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=63 | pages=600–610 | doi=10.2307/2371374 | issue=3}}
*{{Citation | last1=Erdős | first1=Paul | last2=Hajnal | first2=András | title=Axiomatic Set Theory ( Univ. California, Los Angeles, Calif., 1967) | publisher=Amer. Math. Soc. | location=Providence, R.I. | series=Proc. Sympos. Pure Math | mr=0280381 | year=1971 | volume=XIII Part I | chapter=Unsolved problems in set theory | pages=17–48}}
*{{citation|mr=0795592
|last=Erdős|first= Paul|author1-link=Paul Erdős|last2= Hajnal|first2= András|author2-link=András Hajnal|last3= Máté|first3= Attila|last4= Rado|first4= Richard|author4-link=Richard Rado
|title=Combinatorial set theory: partition relations for cardinals
|series=Studies in Logic and the Foundations of Mathematics|volume= 106|publisher= North-Holland Publishing Co., |place=Amsterdam|year= 1984| isbn= 0-444-86157-2 }}
*{{citation|mr=0081864
|last=Erdős|first= P.|author1-link=Paul Erdős|last2= Rado|first2= R.|author2-link=Richard Rado
|title=A partition calculus in set theory
|journal=Bull. Amer. Math. Soc. |volume=62 |year=1956|pages= 427–489
|url=http://www.ams.org/bull/1956-62-05/S0002-9904-1956-10036-0/|doi=10.1090/S0002-9904-1956-10036-0|issue=5}}
* {{cite book|author=Kanamori, Akihiro|authorlink=Akihiro Kanamori|title=The Higher Infinite, second edition|publisher=Springer|year=2000|isbn=3-540-00384-3}}
*{{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}}
 
==Notes==
 
<references/>
 
[[Category:Set theory|*]]
[[Category:Combinatorics|*]]

Latest revision as of 05:43, 21 February 2014


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