Infinitary combinatorics

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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

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Write   for ordinals,   for a cardinal number (finite or infinite) and   for a natural number. Erdős & Rado (1956) introduced the notation

 

as a shorthand way of saying that every partition of the set   of  -element subsets of   into   pieces has a homogeneous set of order type  . A homogeneous set is in this case a subset of   such that every  -element subset is in the same element of the partition. When   is 2 it is often omitted. Such statements are known as partition relations.

Assuming the axiom of choice, there are no ordinals   with  , so   is usually taken to be finite. An extension where   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   into   pieces has a subset of order type   such that for any finite  , all subsets of size   are in the same element of the partition. When   is 2 it is often omitted.

Another variation is the notation

 

which is a shorthand way of saying that every coloring of the set   of  -element subsets of   with 2 colors has a subset of order type   such that all elements of   have the first color, or a subset of order type   such that all elements of   have the second color.

Some properties of this include: (in what follows   is a cardinal)

  for all finite   and   (Ramsey's theorem).
  (the Erdős–Rado theorem.)
  (the Sierpiński theorem)
 
  (the 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

 

Strong colorings

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Wacław Sierpiński showed that the Ramsey theorem does not extend to sets of size  by showing that  . That is, Sierpiński constructed a coloring of pairs of real numbers into two colors such that for every uncountable subset of real numbers  ,   takes both colors. Taking any set of real numbers of size   and applying the coloring of Sierpiński to it, we get that  . Colorings such as this are known as strong colorings[3] and studied in set theory. Erdős, Hajnal & Rado (1965) introduced a similar notation as above for this.

Write   for ordinals,   for a cardinal number (finite or infinite) and   for a natural number. Then

 

is a shorthand way of saying that there exists a coloring of the set   of  -element subsets of   into   pieces such that every set of order type   is a rainbow set. A rainbow set is in this case a subset   of   such that   takes all   colors. When   is 2 it is often omitted. Such statements are known as negative square bracket partition relations.

Another variation is the notation

 

which is a shorthand way of saying that there exists a coloring of the set   of 2-element subsets of   with   colors such that for every subset   of order type   and every subset   of order type  , the set   takes all   colors.

Some properties of this include: (in what follows   is a cardinal)

  (Sierpiński)
  (Sierpiński)
  (Laver, Blass)
  ( Galvin and Shelah)
  (Todorčević)
  (Moore)
  ( Galvin and Shelah)

Large cardinals

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Several large cardinal properties can be defined using this notation. In particular:

  • Weakly compact cardinals   are those that satisfy  
  • α-Erdős cardinals   are the smallest that satisfy  
  • Ramsey cardinals   are those that satisfy  

Notes

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  1. ^ Andreas Blass, Combinatorial Cardinal Characteristics of the Continuum, Chapter 6 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010
  2. ^ Todd Eisworth, Successors of Singular Cardinals Chapter 15 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010
  3. ^ Rinot, Assaf, Tutorial on strong colorings and their applications, 6th European Set Theory Conference, retrieved 2023-12-10

References

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