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.
for a cardinal number (finite or infinite) and
Erdős & Rado (1956) introduced the notation as a shorthand way of saying that every partition of the set
pieces has a homogeneous set of order type
A homogeneous set is in this case a subset of
-element subset is in the same element of the partition.
Such statements are known as partition relations.
Assuming the axiom of choice, there are no ordinals
κ → ( ω
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
pieces has a subset of order type
Another variation is the notation which is a shorthand way of saying that every coloring of the set
with 2 colors has a subset of order type
have the first color, or a subset of order type
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 Wacław Sierpiński showed that the Ramsey theorem does not extend to sets of size
That is, Sierpiński constructed a coloring of pairs of real numbers into two colors such that for every uncountable subset of real numbers
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.
for a cardinal number (finite or infinite) and
Then is a shorthand way of saying that there exists a coloring of the set
pieces such that every set of order type
A rainbow set is in this case a subset
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
is a cardinal) Several large cardinal properties can be defined using this notation.