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 and combinatorics on successors of singular cardinals.

Ramsey theory for infinite sets

for a cardinal number for a natural number. introduced the notation as a shorthand way of saying that every partition-element subsets is 2 it is often omitted. Such statements are known as partition relations.

Assuming the axiom of choice is almost allowed to be infinite is the notation which is a shorthand way of saying that every partition is 2 it is often omitted.

Another variation is the notation .

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. That is, Sierpiński constructed a coloring of pairs of real numbers. Colorings such as this are known as strong colorings and studied in set theory. introduced a similar notation as above for this.

for a natural number. Then is 2 it is often omitted. Such statements are known as negative square bracket partition relations.

Another variation is the notation

colors.

is a cardinal)

Large cardinals

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

• Weakly compact cardinals

• α-Erdős cardinals

• Ramsey cardinals

Notes

References