Infinitary combinatorics explained

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

\kappa, λ for ordinals, m for a cardinal number (finite or infinite) and n for a natural number. introduced the notationas a shorthand way of saying that every partition of the set [\kappa] ^{n} of n -element subsets of \kappa into m pieces has a homogeneous set of order type λ . A homogeneous set is in this case a subset of \kappa such that every n -element subset is in the same element of the partition. When m is 2 it is often omitted. Such statements are known as partition relations.

Assuming the axiom of choice, there are no ordinals

\kappa with \kappa → (\omega) ^{\omega} , so n is usually taken to be finite. An extension where n is almost allowed to be infinite is the notationwhich is a shorthand way of saying that every partition of the set of finite subsets of \kappa 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 notationwhich is a shorthand way of saying that every coloring of the set

[\kappa] ^{n} of n -element subsets of \kappa 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 \mu such that all elements of [\mu] ^{n} have the second color.

Some properties of this include: (in what follows

\kappa is a cardinal)

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

Wacław Sierpiński showed that the Ramsey theorem does not extend to sets of size

\alef _{1} by showing that 2 ^{\alef _{0}} \nrightarrow(\alef ^{2} _{2} . That is, Sierpiński constructed a coloring of pairs of real numbers into two colors such that for every uncountable subset of real numbers X , [X] ^{2} takes both colors. Taking any set of real numbers of size \alef _{1} and applying the coloring of Sierpiński to it, we get that \alef _{1\not → (\alef} ^{2} ^{2} . Colorings such as this are known as strong colorings and studied in set theory. introduced a similar notation as above for this.

Write

\kappa, λ for ordinals, m for a cardinal number (finite or infinite) and n for a natural number. Thenis a shorthand way of saying that there exists a coloring of the set [\kappa] ^{n} of n -element subsets of \kappa into m pieces such that every set of order type λ is a rainbow set. A rainbow set is in this case a subset A of \kappa such that [A] ^{n} takes all m colors. When m is 2 it is often omitted. Such statements are known as negative square bracket partition relations.

Another variation is the notation

\kappa\nrightarrow[ λ ; \mu] ^{2} _{m} which is a shorthand way of saying that there exists a coloring of the set [\kappa] ^{2} of 2-element subsets of \kappa with m colors such that for every subset A of order type λ and every subset B of order type \mu , the set A x B takes all m colors.

Some properties of this include: (in what follows

\kappa is a cardinal)

Large cardinals

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

• Weakly compact cardinals

\kappa are those that satisfy \kappa → (\kappa) ^{2}

• α-Erdős cardinals

\kappa are the smallest that satisfy \kappa → (\alpha) ^{<\omega}

• Ramsey cardinals

\kappa are those that satisfy \kappa → (\kappa) ^{<\omega}

Notes

1. [Andreas Blass]
2. Todd Eisworth, Successors of Singular Cardinals Chapter 15 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010