It seems that these three tags are being discussed in this thread: infinite-combinatorics, small-uncountable-cardinals, cardinal-characteristics.The tag infinite-combinatorics exists for a long time, but it might be good to improve the tag-info. Whether or not the other two tags should be created is part of this discussion - again, clarifying the content of the tag and suggesting the tag-info might be useful in deciding whether the tag might useful.

Let us use this community-wiki answer to write down some suggestions for the tag-info. (And in the comments we can discuss what should be included.) The post is made CW explicitly to encourage editing by other users. (After all, this is the purpose why the community-wiki feature exists.) So do not hesitate to edit the proposed tag-infos if you have any improvements or additions.

**The tag (infinite-combinatorics)**

This tag currently has a short tag-excerpt and empty tag-wiki.

Infinite combinatorics deals with various combinatorial properties of infinite sets. The topics might include, for example,

- Ramsey theory on countably infinite sets, including results related to Szemerédi's theorem, Hindman's theorem, etc.
- Ramsey theory on uncountable sets, such as the Erdős–Rado theorem, and partition calculus
- Diamond ($\diamondsuit$) principles and relatives (such as $\clubsuit$), square ($\Box$) principles, club-guessing principles
- Combinatorial properties of infinite graphs or partial orders (such as their chromatic number, marriage problems, etc)
- Cardinal characteristic of the continuum and related topics
- Infinite trees, such as Kurepa trees or Aronszajn trees
- Ramsey ultrafilters, p-points and related topics
- (Maximal) almost disjoint families

It may be desirable to rename this (infinitary-combinatorics). The tag exists in math.stackexchange with excerpt

For topics of a combinatorial character in set theory. Topics belonging to "combinatorial set theory" may be tagged this way. These include: Partition calculus, diamond principles, square principles, combinatorial properties of infinite graphs or partial orders, etc.

and tag info

This tag is for topics of a combinatorial character studied in set theory. Topics belonging to "combinatorial set theory" or "infinitary combinatorics" may be tagged this way. These include: Partition calculus (generalizations of Ramsey theory to infinite cardinals, infinite ordinals, other partially ordered structures, etc), diamond ($\diamondsuit$) principles and relatives (such as $\clubsuit$), square ($\Box$) principles, club-guessing principles, combinatorial properties of infinite graphs or partial orders (such as their chromatic number, marriage problems, etc), among others.

(The name infinite-combinatorics was briefly used over there when the tag was first being discussed, simply due to the limitation on the number of characters on tag names that we used to have. We switched to infinitary-combinatorics when we saw that limitation was no longer in effect.)

**The tag (small-uncountable-cardinals)**

The tag currently has a short tag excerpt and empty tag-info. (The tag was created very recently - it is possible that, depending on the outcome of this discussion, it might be either removed or possibly moderators might rename this tag.)

Small uncountable cardinals or cardinal characteristics of continuum are various cardinals which are typically between $\aleph_1$ and $2^{\aleph_0}$ and their definition often has a combinatorial flavor. Some examples are:

- The cardinal $\mathfrak p$ - the smallest cardinality of subsystem of $[\omega]^\omega$ with strong finite intersection property and no pseudointersection.
- Various cardinals related to $(\omega^\omega,\le^*)$ such as the bounding number $\mathfrak b$ (=the smallest cardinality of an unbounded subset) or the dominating number $\mathfrak d$ (=the smallest cardinality of dominating subset).

See also: Cardinal characteristic of the continuum on Wikipedia.

**The tag (cardinal-characteristics)**

A suggestion for the tag-info:

Cardinal characteristics of continuum are various cardinals which are typically between $\aleph_1$ and $2^{\aleph_0}$ and their definition often has a combinatorial flavor. Some examples are:

- The cardinal $\mathfrak p$ - the smallest cardinality of subsystem of $[\omega]^\omega$ with strong finite intersection property and no pseudointersection.
- Various cardinals related to $(\omega^\omega,\le^*)$ such as the bounding number $\mathfrak b$ (=the smallest cardinality of an unbounded subset) or the dominating number $\mathfrak d$ (=the smallest cardinality of dominating subset).

See also: Cardinal characteristic of the continuum on Wikipedia.

The tag also encompasses analogues at larger cardinals, such as the bounding number $\mathfrak b(\kappa)$ defined in terms of families of functions from $\kappa$ to $\kappa$, and cardinal invariants of certain structures (such as topological spaces or Boolean algebras).

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