# [Combinatorics]

It seems to me that problems in infinitary combinatorics (a significant area in set theory) should naturally be tagged "". Some users disagree and remove the tag from such questions.

Should the description of the tag "combinatorics" be expanded so the inclusion of these questions is clear, should a new "infinitary combinatorics" tag be created, or do you see a better alternative?

• I know this field under the name of "combinatorial set theory" (presuming you're talking about Ramsey theory and the like). – Lord_Farin Jul 27 '13 at 21:20
• I felt that until some time ago, that could have easily been tagged under [set-theory] and/or [cardinals]. Perhaps it's time to add another set theoretic tag. – Asaf Karagila Jul 27 '13 at 22:49
• I think that a tag infinite-combinatorics, covering partition calculus and the various branches of infinite Ramsey theory would be a fine idea, leaving combinatorics for finite combinatorics. I glance at everything on the Questions page, but I’d actually follow this tag, as I do general-topology (among a few others). – Brian M. Scott Jul 28 '13 at 9:08

Brian's suggestion, of a tag seems the best solution to avoid clashes.

Here is what I wrote for the tag description, feel free to go ahead and make appropriate changes:

Excerpt:

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

Description:

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.

• You should write a tag wiki/excerpt. – Asaf Karagila Jul 28 '13 at 15:06
• @AsafKaragila Done. – Andrés E. Caicedo Jul 28 '13 at 15:35
• By the way, $\LaTeX$ works in the excerpt now. So you can use $\lozenge$ and $\square$ and so on there as well. – Asaf Karagila Jul 28 '13 at 15:42
• [Infinitary-combinatorics] seems less likely to be confused with combinatorial questions on formal power series ("how to prove Euler's pentagonal theorem?"), and more accurate/traditional linguistically. I'd also add [partition-calculus] as a synonym (which, oddly enough, is another name that can be confused with combinatorial identities between formal power series that enumerate partitions). – zyx Jul 28 '13 at 16:55
• @zyx I agree, and I've asked the moderator to change the name, didn't see how to do that myself. On the other hand, I do not think that [partition-calculus] should be a synonym (formally, it is not). If anything, it should be another tag, more specific than [infinitary-combinatorics], but I do not want to add it yet, unless it becomes clear that there is need for such specificity. – Andrés E. Caicedo Jul 28 '13 at 18:44
• I agree that infinitary combinatorics should be better. But I also don't see how to avoid the possible mis-tagging of some questions. Then again, there are people posting seemingly arbitrary questions as [logic] and [set-theory], so what's one more tag? :-) – Asaf Karagila Jul 28 '13 at 18:59
• I agree that mistagging will be unavoidable, no matter what we do. Anyway, I do not expect it shouldn't be much of a problem, but more importantly, I do not think that names should be chosen with the goal of minimizing mistagging. – Andrés E. Caicedo Jul 28 '13 at 19:11
• Is $\square$ usually read square nowadays? When I was in grad school at Madison it was always box. – Brian M. Scott Jul 28 '13 at 19:30
• @BrianM.Scott It has always been square for me. It seems to be the accepted current name. – Andrés E. Caicedo Jul 28 '13 at 19:40
• Thanks. It was ~$40$ years ago, and I suppose that it could also have been a local thing. – Brian M. Scott Jul 28 '13 at 19:42
• @BrianM.Scott Several people (Burke comes to mind) still use Box, though. One of those things, as the direction of the ordering in forcing. – Andrés E. Caicedo Jul 28 '13 at 19:48
• @Brian: Interesting. I never knew it had a different name! – Asaf Karagila Jul 28 '13 at 19:50
• In my experience it is "square" in set theory and "box" in modal logic. – Carl Mummert Jul 29 '13 at 1:50
• @Andres: my answer is somewhat a response to this proposal – Carl Mummert Jul 29 '13 at 2:08
• @zyx: I've renamed the tag as you suggested. – Willie Wong Jul 29 '13 at 7:51

There are various results in infinite combinatorics that are not usually considered "set theoretic combinatorics", such as

• Hindman's theorem
• Szemeredi's theroem
• The Carlson-Simpson theorem
• The Green-Tao theorem

I don't see any problem with labeling these "combinatorics", and I hope people are not removing the tag from them.

On the other hand, I would be suprised to find a question on Martin's axiom among questions tagged "combinatorics", but also (somewhat) surprised to find it among "infinitary combinatorics" questions. I would call it "set-theoretic combinatorics".

So I am not sure that just adding an "infinitary combinatorics" tag will resolve the situation. There are really at least three different classes of combinatorics questions, which I can roughly characterize by typical examples:

1. Finite Ramsey's theorem / Four coloring theorem / Generating functions (finite combinatorics)
2. Hindman's theorem / Szemeredi's theorem / Green-Tao Theorem (infinite combinatorics, but no set-theoretic issues)
3. Martin's axiom / Jensen's diamond / Cardinal invariants of the continuum (infinite combinatorics with genuine set-theoretic issues)

Having two tags will force two of these to be lumped together. I would prefer to combine the first two rather than 2 and 3. There are also more difficult cases, such as the Erdos-Rado theorem.

• (Hindman's theorem falls naturally under set theoretic Ramsey theory, so it would be natural to tag it infinitary-combinatorics. But I get the point.) That said, I see no issues with a question being tagged Ramsey theory, combinatorics, infinitary-combinatorics (and others) simultaneously. At worst, different audiences may look at it and one would get a larger amount of perspectives than usual. – Andrés E. Caicedo Jul 29 '13 at 2:11
• The issue with Hindman's theorem, of course, is that it has no set-theoretic issues (it's even provable in modest systems of second-order arithmetic). So it is not a great match for a tag wiki that refers to topics "studied in set theory". I would speculate most interest in Hindman's theorem is outside the field of set theory. The Green-Tao theorem is also of this sort. – Carl Mummert Jul 29 '13 at 2:14
• (I would disagree, but this is really a matter of opinion, and off-topic.) To see if I understand, would you think perhaps a new tag is in order? Say "infinite combinatorics" for the Ramsey theoretic results you list in 2, and "infinitary combinatorics" for the ones you list in 3. (Roughly, of course, these lines are blurry and all that.) – Andrés E. Caicedo Jul 29 '13 at 2:42
• The one comment I have in that regard is that the Ramsey-theory tag does not have many entries, so I am not sure whether currently there is need for this one, which would essentially be a sub-tag. (But, of course, it is something that I see is potentially useful.) – Andrés E. Caicedo Jul 29 '13 at 2:44